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The prime minister is also ready to endorse the chancellor unequivocally if he can be assured Brown is fully signed up to the reform agenda that was agreed by the cabinet last month.
Until now Blair believed it would be better for the chancellor to face a challenge that would force him to spell out his policies.
However, the prime minister has been speculating privately to aides that an uncontested “coronation” for Brown would be best, as a contest could be divisive and ruin any chances of a smooth transition of power.
His decision may also have been prompted by practicalities. In the past fortnight it has emerged that no cabinet colleague is serious about challenging Brown, who looks a shoo-in for the top job.
He may face a minor challenge from a left-wing Labour MP such as John McDonnell or Michael Meacher, but they may struggle to get enough signatories to back their bid.
“Tony accepts that Gordon is the natural successor and thinks he deserves it, but some days he still has a few doubts,” said a No 10 insider. “If Gordon helps himself, then we have to help him. He needs to lighten up, distance himself from the backstabbers and show he can be collegiate.”
Insiders say Blair sometimes expresses “guilt” about breaking his past pacts with Brown to give up office; but at other times he has talked about his wish to “skip a generation” by persuading David Miliband, the young and articulate environment secretary, to go for the premiership.
The nagging doubts Blair has about the succession were demonstrated last week. He came close to endorsing Brown, something the chancellor has long sought, when he chided David Cameron, saying that the Tories would face the “big clunking fist” of a “heavyweight” at the next general election.
However, Blair and his spokesman later cast doubt on whether the comments amounted to an endorsement by sidestepping questions on the subject. John Reid and John Hutton, both Blair loyalists, were also sceptical.
Yesterday Blair and Brown were playing international statesmen with trips abroad.
The chancellor met troops in Basra on his first visit to Iraq, praising their “tremendous morale” and the “work they have been doing in often very difficult circumstances”. He promised an extra £100m over three years to help to rebuild the country’s economy.
Earlier Blair, who arrived in Pakistan yesterday for talks, appeared to agree that violence in Iraq since the 2003 invasion had been a “disaster”. He announced £480m in aid for moderate Muslim schools in Pakistan and a new cooperation deal between the two countries’ intelligence services.
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