Others have been rather less coy. Lord Archer of Weston-super-Mare, the Conservative novelist and millionaire, has been sending his unofficial manifesto to chairs of governors of London schools, and Labour has two declared candidates: Ken Livingstone, the front-runner, and the sports minister Tony Banks.
All this is for a post that does not yet exist. On Thursday, local election day, Londoners are expected to back the establishment of a mayor and a London authority in a referendum, freeing the parties to choose their candidates.
He or she will be a powerful figure with a big personal mandate, access to the London-based media and powers over transport, the environment, tourism and inward investment. That is why both party leaderships want alternatives to their front-runners, Mr Livingstone and Lord Archer. On the Conservative side Lord Archer remains the activists' but not the public's choice.
In an NOP/Evening Standard poll, 55 per cent said they would "definitely" not vote for him, although Lord Archer is still more popular than his party. Last week the peer withdrew from a televised debate at the last moment, prompting some excited but unfounded speculation that he might be poised to withdraw from the contest.
Potential alternatives like Chris Patten, ex-governor of Hong Kong (who has said he has no intention of standing); Sebastian Coe, the runner and ex-Tory MP; and Steve Norris, the former minister, will be concerned to watch how the party bears up in the local elections. After all, why get into a difficult scrap to win the nomination if the Tory is destined to lose?
The Tories ought to make gains because they are defending about 500 seats they won in 1994 - a bad year for them. Labour, which has specialised in urging supporters to use local elections to punish previous Tory governments, is struggling to motivate its backers. Local council by-elections have not been good. Nevertheless, there is anxiety in Conservative Central Office based on the party's low opinion poll rating. In London the task is harder still because activists have left in droves, discouraged by last year's general election clear out.
On the Labour side, the calculation is more complicated. The party hierarchy is divided between those who think that Mr Livingstone should be denied the official candidature through (yet to be constructed) party machinery, and those who believe this would provoke a backlash. This decision is one of Mr Blair's most acute political dilemmas because Mr Livingstone could become a running sore as the next general election approaches. Downing Street is delaying the selection until next year, although it looks likely that the National Executive Committee will have the power to block "unsuitable" candidates.
Whether or not this is necessary will depend on a viable alternative to Mr Livingstone emerging. Hence the leadership's contentment with Ms Jackson's moves. Her showing in last week's poll - 47 per cent to Mr Livingstone's 55 per cent - is very respectable.
Other individuals may come forward with two Cabinet ministers, Chris Smith and Frank Dobson, mentioned (despite their denials). Here, there is an element of chicken and egg because no Cabinet minister will want to take on the former GLC leader in a ballot of all members knowing defeat is a likely outcome. Thus the elimination of Mr Livingstone may be the key to others coming forward.
That puts the ball squarely back in Mr Blair's court. One party official argues: "He will have to confront the fact that, if Ken Livingstone were to become mayor, it would be the biggest daily problem faced by Tony Blair and his government. Constant criticism from that quarter would be more significant in the eyes of journalists and broadcasters than anything that could be said by William Hague or Paddy Ashdown."