Mr Cameron had lunch at Chequers with his old friend and rival this week.
He is then to travel to London that evening, to have dinner with Prime Minister Tony Blair at Chequers.
The talks at Chequers will focus on cross-border security and how to engage the Taliban in effective peace talks.
The first hot spot was Chequers Inn in Doddington, once a stopping-point for religious pilgrims en route to nearby Canterbury.
German Chancellor Angela Merkel is due to have talks with the British prime minister at his country retreat of Chequers on Saturday.
Rodney Melville, formerly secretary at the Chequers Trust, for services to architecture.
The newspaper says David Cameron and Andrew Mitchell had lunch at Chequers with their two wives, after which Mr Cameron made the job offer.
Mr Cameron hosted a dinner with Afghan President Hamid Karzai and Pakistani President Asif Ali Zardari at Chequers, north of London, on Sunday evening.
Nevertheless, after some resistance, Mr Cameron has published a list of all big Conservative donors who have dined with him in Downing Street or Chequers, his official rural retreat.
It was reported earlier this year that Mr Cameron had sought the former Labour prime minister's advice since taking power, mainly on foreign issues, and invited him to Chequers.
In future, Mr Cameron said the Conservatives would publish details of all meals with donors, whether at No 10 or Chequers, as well as a register of all meetings with those who have given money.
BBC: David Cameron publishes list of all donors dining at No 10
It said it wanted the names of Tory donors who have visited government property - including Downing Street and Chequers - and of those who have made submissions to the Downing Street policy unit, to be published.
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