When Mr Netanyahu was elected, to the horror of the Israeli media, in 1996, his victory was attributed to the Rasputin-like skills of Arthur Finkelstein, a veteran right-wing Republican consultant. This time around Mr Barak has the services of three Democratic advisers: James Carville, Stanley Greenberg and Robert Shrum. Probably their greatest achievement so far has been to give confidence to One Israel, formerly called the Labour Party and still badly rattled by its defeat three years ago, that it can win against Mr Netanyahu.
In the past month Mr Barak's campaign has come together. He has successfully avoided television debates, where he performs poorly, and he looks less like a military officer who has wandered into politics through mistaken ambition. The party leadership has united behind him: unlike last year, he is zig-zagging adeptly between secular and religious interests, which are the deepest divide in Israeli politics.
Just how much Mr Carville has to do with this is dubious. In interviews with Israeli journalists he seems touchingly ignorant of most of the issues in Israeli politics, though he has improved. He has stopped referring to Mr Barak as "general" - a title unlikely to cut much ice in a country awash with senior officers who have entered politics, and unlikely to endear One Israel to Israeli Arabs, who must turn out en masse if he is to win.
The problem for the American consultants is that - for all the talk of the Americanisation of Israel - Israeli politics are very different from those of the US. Traditionally campaigns have made only a limited difference to the outcome of elections. Secular and religious loyalties, reinforced by ethnic background, run too deep.
Ironically, the Democratic consultants are working for a party which resembles the US Republicans of the 1950s. One Israel is the party of the establishment, the founders of Israel, the better-off, the intelligentsia, secular in tradition and of Ashkenazi origin. Like the Republicans of 40 years ago, it has difficulty in expanding beyond a particular core of support.
Like the Democrats in the US, that situation has enabled the so-called Israeli right to assemble a coalition of Israelis who feel excluded from power. They include a veteran right-wing leadership, the ultra-orthodox, the religious, Jews whose families come originally from North Africa or the Middle East, the poorer towns and immigrants from the former Soviet Union. Many of these elements have nothing in common, but they share a sense of deprivation.
They also form a majority. Labour held power from the foundation of the state of Israel in 1948 up to 1977, but has won only one election since. The right-wing coalition is estimated to make up 53 per cent of the population. Class, ethnic background and religion reinforce each other and divisions are deep. Elections in Israel usually see voters return to the fold of their own communities because of peer pressure, rather than striking out in new directions.
The strength of traditional loyalties was shown after the assassination of Yitzhak Rabin, the Labour Prime Minister, in 1995. It convulsed Israeli politics, but it did not change allegiances. Nevertheless, there are divisions within Mr Netanyahu's camp. The biggest mistake by the last Labour government was to ignore the million Russian immigrants, the fastest growing and politically most fluid sector of the six million Israeli population. In 1992 they voted for Mr Rabin. In 1996 they switched to Mr Netanyahu. Their main motive in both cases was that they felt ignored by the government.
Although Russian Jews were far better treated than were Jewish immigrants from the Middle East in the 1950s, their problem is that they are overwhelmingly white-collar. No fewer than 78,000 of the Russians were engineers before they arrived in Israel and 40,000 were teachers, but only a small proportion - some 6,000 in the teachers' case - could find similar work in their new home.
Mr Finkelstein says that the key to Mr Netanyahu's victory is to increase his proportion of the Russian vote, but that is not happening. In the past month polls show Mr Barak's proportion of the Russian vote rising from 19 per cent to 30 per cent in a run-off against Mr Netanyahu on 1 June. He has astutely all but offered Yisrael Ba'aliya, the main Russian party, the Interior Ministry in the next government. That matters to the Russians, because it controls the right of residence in Israel.
Those moves put Mr Netanyahu in a trap. The Interior Ministry has long been the fief of Shas, the third largest party, whose supporters are mainly religious Jews from North Africa and the Middle East. Aryeh Deri, the leader, came out in support of Mr Netanyahu last week, so the Prime Minister cannot top Mr Barak's offer to the Russians.
Coalition-building, not electoral campaigning, is at the heart of Israeli politics; it is that which won Mr Netanyahu the last election. But even so, the right and left are evenly balanced. In 1992, the one Labour Party victory in a quarter of a century, the right actually had a majority of 11,000 votes but lost because some of those votes were wasted on small parties. In the last election Mr Netanyahu won by 30,000 votes out of three million. Despite the fabled skills of Mr Finkelstein, the right did far better at the ballot box than the Prime Minister.
The real contribution of American consultants may be psychological rather than real. In Israel they have celebrity status, so their very presence boosts their party's morale and dismays the opposition. They enable both Mr Netanyahu and Mr Barak to take tactical day-to-day decisions out of the hands of wrangling party barons - one of Labour's weaknesses three years ago.
Whatever happens, either Mr Finkelstein or Mr Carville will be able to claim credit for a famous victory.