For Britain, getting to the head of the queue was vital from the moment that Mr Clinton was elected - to show that the new White House still has a special place in its affections for the Old Country and that stories suggesting an early animosity between the two leaders are unfounded.
For the President, though, the Prime Minister's visit may be little more than a distraction from his almost all-consuming task of selling his domestic economic proposals to Congress and voters. But then he will only see Mr Major for an hour tomorrow afternoon and later for a 'working supper'.
Even during that time, Mr Major may struggle to get the President's full attention. The month-old administration has showed scant sign of thinking much about Europe, let alone Britain. Nor are we certain that Mr Clinton does not hold some lingering grudge against Mr Major for the help given to the Bush re-election campaign by the Conservative Party.
The impression has also grown in Washington, as elsewhere, that Britain, under Mr Major's leadership, is on a path of decline that is taking it ever further from the centre of world affairs, both politically and economically. From this arises the concern that where Britain once offered the United States a useful conduit into the European Community, it may not any more. 'From this angle, the place looks really, really off-track,' said Mark Nelson of the Carnegie Foundation. 'And especially off-track with regard to Europe. If Britain loses that perceived influence over European affairs then it will find America won't give a hoot about it any more.'
And as the Clinton administration casts its foreign policy view in an economic, rather than security, context, it may be that the marginalisation of Britain, for the US, is anyway unavoidable. 'Without any bad feeling intended, there is going to be a dramatic change in US-UK relations,' said Bill Maynes, editor of Foreign Policy magazine in Washington. 'As long as security was the main interest, Britain was always going to play a disproportionately large role. With the economy rising to the top, Britain's role recedes.'
The perception of Britain and Mr Major sliding into some sort of oblivion is reinforced by reports from American correspondents in London. The Washington Post's reporter noted yesterday, for example, that Mr Major 'seems to have an uncanny instinct for making a beeline towards the brink of political disaster'.
There are, however, some substantive issues - including security ones - to which the two leaders can usefully turn their attention. Top of the list, inevitably, will be Bosnia and Mr Clinton's intentions with regard to the Cyrus Vance-Lord Owen peace plan.
The natural bent of both men towards easy chumminess will probably ensure that this first meeting will go off quite happily, with both sides reassured that all is well between Britain and America. In substance, little may be achieved and it could be that the hallowed 'special relationship' is left quietly to fade away.