The poll was carried out on Thursday - after the Government's embarrassing retreat in its latest battle with its European Union partners, and after the call from Tony Marlow, the backbench Tory MP, for Mr Major to resign. The poll shows that 58 per cent of all voters (and 39 per cent of Tory voters) think Mr Major handled the latest dispute with the EU quite badly or very badly; only 19 per cent think he handled it well.
The Tories' best bet, according to the poll, is to recall Margaret Thatcher. Thirty-one per cent say they would vote Tory in a general election if she were leader, compared with 26 per cent if John Major continues. But Baroness Thatcher's return is almost an impossibility. In any case, Michael Heseltine would do almost as well, giving the Tories 30 per cent of the vote. If Douglas Hurd or Kenneth Clarke were elected leader, support would drop to 24 per cent.
Mr Heseltine's position, however, is nothing like as strong as it was in 1990, the year of Lady Thatcher's overthrow. Then, the polls suggested that he could make the difference between victory and a heavy defeat. In November 1990, the polls said that if Mr Heseltine took over from Lady Thatcher, the Tories would enjoy a 7 per cent swing. This time, the figures suggest he would do no more than help to salvage something from a catastrophe.
The poll also suggests that Mr Major's personal plight is not as bad as Lady Thatcher's was in 1990. Several polls in the spring of that year showed more than 60 per cent of voters wanting her to step down; Mr Major is still well short of that level of unpopularity. But the Tory party's position is worse than in 1990. In that year, the polls persistently indicated that the Conservatives would retain more than 30 per cent of an election vote. Most polls this year, however, have put the Tory vote at well under 30 per cent.
There are other reasons for thinking that the Tories have nothing to gain from ditching Mr Major. NOP asked respondents to give the two factors (from a choice of nine) that were most important to them in deciding how to vote. The economy came out easily top, named by 54 per cent. It was followed by 'the level of taxes, such as income tax and VAT' (45 per cent) and policies on law and order (33 per cent).
All other factors came well behind. The Government's policies on Europe, for example, were chosen by 14 per cent. 'The quality of the person leading the Conservative Party' was nominated by only 12 per cent. Rather more - 18 per cent - said they just wanted 'to get rid of the Conservatives'.
Significantly, the leadership was a much more important factor among Tory voters - 22 per cent chose it, against only 5 per cent of Labour voters and 11 per cent of Liberal Democrat voters. In other words, by ousting Mr Major the Tories could upset some of their core supporters without gaining much from the other parties.
But there are several reasons for believing that the tide is running in Mr Heseltine's favour. First, his position is much improved on that shown by an ICM poll in the Guardian six weeks ago. ICM suggested that electing him leader was as likely to lose votes for the Tories as electing Mr Clarke or Mr Hurd; NOP suggests that he would gain votes, though not many.
Second, Mr Heseltine is particularly strong among a crucial group of voters: those who say they voted Tory in 1992 but would vote Liberal Democrat now. For this group, the Tory leadership is a more important factor than it is for the NOP sample as a whole - nearly a quarter named it when asked to make a choice, putting it fourth in importance behind the economy, taxes and law and order. And 38 per cent of the group say they would go back to the Tories if Heseltine were to become Prime Minister.
A Thatcher leadership would do even better among this group (41 per cent would then vote Tory) but Mr Clarke and Mr Hurd would both do much worse, winning back only 18 and 13 per cent respectively.
The importance of this result for Mr Heseltine's prospects cannot be overestimated. Paddy Ashdown's party is the one that threatens the Tory hegemony in southern England; if Mr Major fails to calm nerves among MPs from these areas, he will not survive.
The third reason for optimism among Heseltine supporters is the tax issue. Voters will not start to see the effects in their pay packets, mortgage repayments and fuel bills until later this month. In 1990, Tory ratings dipped sharply - from the mid-30s to the low 30s - when poll tax bills dropped through letterboxes in March and April. And the higher taxes are likely to damage Mr Clarke.
At the beginning of the year - in the wake of his heart attack last summer - Mr Heseltine looked a political has-been. Now, after subtly distancing himself from the Government's latest disasters, he looks ready to renew his public image, as the party faces heavy defeats in local and European elections and by-elections. In athletics parlance, he may have timed his run to perfection.
NOP interviewed a representative quota sample of 1,135 adults in 53 constituencies across Great Britain on 31 March.