One tries to imagine the scene. Who will raise the subject first? Will they argue the point, or discuss it matter-of-factly as two mature adults? Keating's office will say only that the subject of the conversation will be entirely between monarch and prime minister.
It is certain, however, that Keating will bring the Queen up to date with developments in Australia. A report is expected next month from a body called the Republic Advisory Committee on what Keating has described as 'the minimum constitutional changes necessary to achieve a viable Federal Republic of Australia'. He will also tell the Queen that there will be no change unless a majority of Australians vote for it in a referendum.
Less likely scenarios, on which the Australian media have been speculating, include the suggestion that the Queen may turn the tables on Keating, confide that she has decided to privatise Buckingham Palace, and sound him out on becoming the first president of Great Britain. Such flippancy is likely to be overtaken by more serious attention as the visit draws closer.
How the urbanite Keating, who wears hand-stitched Italian suits and pounds 180 shoes, will fit in with the kilt-and-wellies style of the royals' Scottish summer has prompted wry speculation. 'He doesn't fish, shoot or ride. He doesn't even play golf,' says a Canberra insider. One British newspaper columnist has waggishly suggested that Keating's ownership of a piggery could provide a talking point, but added that he should not pin his hopes on bridge as after-dinner entertainment. He should be prepared instead for a game of charades.
One thing is certain. As they sit down to dinner and stroll in the grounds over the weekend of 18-19 September, the Queen will see a different Paul Keating from the one Australians are used to.
The swaggering, swashbuckling political street-fighter, noted for language that could strip the paint off an outback pub, will be nowhere in sight. Keating will turn on the charm for which he is noted in personal encounters, perhaps as a way of glossing over his awkwardness with royal protocol. This was most blatantly displayed when he lay a guiding hand on the Queen's waist during a reception in Canberra last year, their only meeting since he became prime minister in 1991.
The tabloids had a field day. The Daily Express referred to his 'rude pawing' of the Queen, and printed advice from Sir Les Patterson, the mythical 'Australian cultural attache': 'You have to have a chat with the Queen before you put your hand up her frock.'
If Keating is distracted during his Balmoral visit, it will not be just by his dilemma over how to handle the R-word. Five months after he defied the political odds and won re-election as prime minister in his own right, Keating's command of the domestic political stage has been shattered and the republican issue relegated to the wings.
He has tried to enforce a landmark high court judgment last year, which recognised that Aborigines have native title to own land, by pushing through legislation - but has run into trouble with state leaders, mining companies, farmers and even Aborigines themselves.
His government's budget, delivered last week - with its sobering recipe of cutting the Adollars 16bn ( pounds 7.2bn) deficit by postponing income tax cuts and raising taxes on wine, petrol and other consumer goods - has outraged even more Australians, especially those who voted Labor back for an unprecedented fifth term in March.
'An act of bastardry,' was how Michael Easson, of the New South Wales Labor Council, a union umbrella body, flayed the budget on Thursday. 'Keating bites the bullet of betrayal,' said a headline in the Sydney Morning Herald. A disastrous post-budget opinion poll showed public support at 31 per cent, 23 points behind the opposition Liberal-National coalition and the lowest since Labor came to power a decade ago.
Keating's government lacks a majority in the Senate, the upper house of parliament, where the opposition party is combining with Independent and Green MPs to demand changes to the budget. The last thing Keating wants is another election, which deadlock over the budget would eventually entail.
He has been defending himself the way he knows best, with brute words: the obstructionist Senate MPs were 'unrepresentative swill' and a 'pack of pansies'.
Such language has always been part of Keating's political lexicon, marking him out from other ministers with whom he served in Bob Hawke's four administrations from 1983. But it was only after he ousted Hawke as party leader and prime minister two and a half years ago that it became clear Keating was going to be a leader quite unlike any other in Australia this century, and that he was intent on taking his country on a roller-coaster ride into the future.
That moment came in February last year, with his extraordinary outburst to parliament in Canberra attacking Britain's record in the Pacific during the Second World War, accusing his opponents of 'cultural cringe to a country which decided . . . not to worry about Singapore and not to give our troops back to keep ourselves free from Japanese domination'. No Australian leader had ever indulged in such blatant Pom-bashing. Amid the uproar in Britain, the Sun branded Keating 'The Lizard of Oz'.
Such fervid nationalism on Keating's part was a revelation even to many Australians. It had been suppressed during his eight years as treasurer (or finance minister) under Hawke, a period during which the money markets embraced him for his bold reforms to Australia's previously closeted economy, and Euromoney magazine named him 1984 finance minister of the year.
The dazzling heights of running the country's economy were a long way from Bankstown, the working-class Sydney suburb where he was born in 1944. His parents were third-generation Irish Catholics, and it was from his father,
a boilermaker turned small businessman, that Keating inherited his admiration
for independent spirits and self-made men.
Keating has known no life other than politics. He left school at 15, made crucial contacts among the right wing of New South Wales Young Labor, and won a seat in the federal parliament at 25.
Even to the small circle of friends who know him best, Keating remains a strange amalgam. For all his public bravado, he subscribes to the old-fashioned virtues of family life and personal loyalty. He is devoted to his wife, Annita, a Dutch-born former air hostess, and their four children, and insists on spending his weekends at home. His other obsessions are antiques and classical music (transcending those of his youth in the Sixties, when he managed a Sydney pop group called the Ramrods).
As prime minister, Keating has virtually admitted he has no solution to the highest unemployment in Australia since the Thirties. This is largely a legacy of the recession that followed his term as treasurer. On other fronts, he has proved more liberal than Hawke: he has devoted more attention to women and Aborigines, and abolished the ban on gays in the military. 'I was always on the gays' side,' he is quoted as saying in a leaked transcript of a pre-election talk to his staff. 'As far as the military's concerned, what's a few gays between friends?'
It is Keating the Irish Catholic patriot speaking, with his antipathy towards the old Protestant established order, when he calls on Australians to cut the final constitutional links with Britain and forge a closer identity in the Asia-Pacific region. It is also Keating the seasoned political risk-taker, tapping latent public sentiment and testing how far he can go.
Lately he has toned down the Pom- bashing and dropped talk of changing Australia's flag, which has the Union Jack in the corner. His latest republican speech, in the New South Wales town of Corowa a month ago, was almost one of sweet reasonableness.
'While the British monarch still has our affection and our regard, there is no question that the monarchy commands much less of both. I think this decline has less to do with the problems the royal family has recently faced, than it has to do with changes in Australia, changes in the relationship between Australia and Great Britain, and an understanding in both countries of the different necessities we face.
'The monarchy has had family problems at other times in the past, but Australians did not draw the conclusion that the monarchy had lost its relevance. Today they draw that conclusion because the monarchy is more remote from their lives, and perceived as inappropriate to the sort of nation we must become.'
Defining exactly what that nation is has proved elusive. While opinion polls show that a majority of Australians support Keating's republicanism, some of his critics maintain that his solution of 'minimalist' constitutional change - replacing the Queen with an elected president - is fraught with problems. They argue that he has not thought through the hard questions of how to replace the symbols of monarchy.
Jill Ker Conway, an influential Australian-born author and academic, told a high-profile Sydney audience in an address to a think-tank on Wednesday: 'The minimalist approach is no position at all. I don't believe you can change any symbol in public authority without prompting people eventually to rework the basis of public authority in total. If people do that in a negligent, backhanded or semi-conscious way, the answers will be negative.'
Ms Ker Conway's autobiography, The Road from Corrain, is a best-seller in Australia. Like Keating, she grew up in the wealthy, secure Australia of the Fifties; like Keating, she took up republicanism years before its time. Writing of that period, she asked: 'Why did the crowds go into such an undignified frenzy whenever George III's hard-working but intellectually undistinguished descendants paid a ceremonial visit to Australia? What was wrong with us?'
Forty years later, in a society with far fewer certainties, Keating has had the courage to confront Australians with the same question.