But yesterday, while campaigning in the Eddisbury by-election, she sent the clearest signal that she does not want to move. With the peace process "parked" until the end of summer, she said she wanted to stay in the Province "long enough" to do everything she could to help further the process.
The frankness of her remarks, and their timing, were seen at Westminster as a sign she may be arguing against the job being offered by Mr Blair. Neither does she not want to appear a victim of the Ulster Unionists, following David Trimble's warning to Mr Blair that she has "lost touch" with Unionist opinion.
Peter Mandelson is subtly campaigning for her job in a return to the Cabinet, and he would command Unionist support, although some ministers believe it would be too soon to bring him back after his resignation over his house loan from a fellow minister. Ms Mowlam's deputy, Paul Murphy, is also pressing hard for her portfolio.
Mr Blair will not want his ministers dictating terms to him, after warnings by Frank Dobson, Secretary of State for Health, that he does not want to be pushed into running for the post of Mayor of London.
Ms Mowlam is still likely to leave the Province and yesterday showed why she is such a campaigning asset to Labour, canvassing with gusto in small towns in Cheshire.
She remained upbeat about prospects for rebuilding the peace process. Last night she met Mr Blair to discuss preparations for his meeting today with the former US senator George Mitchell and the Irish Prime Minister, Bertie Ahern.
She said: "People should just remember: the bottle is half-full, not half-empty. There's no party leader come out over the weekend and said `I'm walking'. They all want to find a way forward." Questioned about demands by the Ulster Unionists and Tories for a halt to further releases of paramilitary prisoners until the IRA begins decommissioning its weapons, Ms Mowlam added: "The accelerated release of prisoners is not an easy issue.
"It's part of the Good Friday Agreement. You can't cherry pick. It was disappointing last week but we should keep that in perspective. We should remember how far we have come."
Yesterday the Sinn Fein leader, Gerry Adams, held "focused discussions" with Mr Ahern on the peace bid, saying the party would insist on a time limit for a review of the faltering peace process. He also said Sinn Fein's Ard Comhairle - its policy-directing executive - would meet this week to hear a full report from their Good Friday accord negotiating team on the present status of the peace process in the wake of the Unionists' refusal to give the go-ahead for an executive.
He added: "We should not under-estimate the seriousness of recent developments. We see this as a huge challenge to Tony Blair. And, interestingly enough, it is not from republicans or by Irish nationalists - it is a challenge by `Irish unionists' to the British government."Reuse content