Although Mr Dobson has made it clear that he does not want the job, senior sources in the Government and Labour Party said yesterday he was "wavering" and may now throw his hat into the ring.
"Tony Blair thinks Frank would be an excellent candidate," said one source. "We understand his reservations, but it is difficult to turn a job like this down when the party needs you to do your duty."
Mr Blair's allies insist he would not threaten to dismiss Mr Dobson from his cabinet post if he turned down the request to run for mayor, but would seek to persuade him to enter the race.
A former leader of Camden Council, Mr Dobson, now MP for Holborn and St Pancras, has strong London connections. The Labour leadership looks certain to veto Ken Livingstone, the left-wing MP for Brent East, on the grounds that he has breached a code of conduct for candidates on campaign spending and materials. Mr Dobson is popular among London Labour Party members, and his entry into the race would placate much of the anger at the exclusion of Mr Livingstone.
Mr Dobson, who was 59 yesterday, has argued that the job should go to a younger man or woman. He has told friends he does not want to become "Mr London" because he loves his job at the Department of Health and does not want to subject his wife and three children to the intense media scrutiny that running for mayor would provoke.
Trevor Phillips, the broadcaster, said on Sunday that he would run for mayor and hoped to stand as a Labour candidate.