Keir Starmer has met Andy Burnham for the first time since the Makerfield byelection in what sources said was a “frosty” meeting to thrash out a transition of power.

The prime minister has agreed for his likely successor to have talks with the civil service to smooth his path, but there is deep resentment within his inner circle towards Burnham for ousting Starmer.

One Downing Street insider suggested that tensions at the meeting – which was held away from No 10 and is understood to have lasted around an hour – were indicative of the strained relationship between the two sides.

“Keir has shown time and again that he will put the interests of the country first but it is fair to say the meeting wasn’t the warmest. Frosty, even,” they added.

Allies of Starmer said that although he was serious about an orderly handover, he had no qualms in denying Burnham – who had initially hoped to take over in September – a long coronation in order to prepare for government.

“There was a strong push from the Burnham camp to be given longer. But why should they tell Keir they want him out, then expect him to manage the ship through a potentially difficult summer? Keir will of course cooperate on transition, but it will be through gritted teeth,” one said.

Some in Burnham’s team were exasperated about the shorter timetable. “The last lot had years to prepare and still fucked it up. We’ll just have to do it in three weeks,” one senior source said. “The length of the transition will focus minds.”

Starmer has agreed that his likely successor can have access talks with the civil service, including the cabinet secretary, Antonia Romeo, before he enters Downing Street. The process normally takes place with the leader of the opposition before a general election.

He also told the cabinet he wants to “resolve difficult issues” before he leaves No 10. In an indication of his determination to publish the defence investment plan before being replaced, he said it was his job to resolve outstanding problems in his remaining weeks in office.

However, Starmer’s spokesperson said he had agreed there would be “no new major policy or spending commitments” before he stands down that could tie his successor’s hands. This did not include the defence plan, which the prime minister regards as settled.

Sources in Burnham’s camp said he accepted Starmer’s defence spending plan would go ahead, despite reports he would prefer to make the final decision himself. One ally said it was right for the prime minister to “grip” the issue.

However, some Labour backbenchers remain sceptical about Burnham. “He’s totally underprepared and heading straight into a firestorm,” one said. Another said he was “seriously underpowered” on foreign policy and defence.

His allies point to his experience as a cabinet minister – at the treasury and the culture and health departments – as well as his nine years as Greater Manchester mayor, and say transition and policy teams have been operating for weeks.

Louise Haigh, who ran his successful byelection campaign, is overseeing the transition for him, alongside Ed Miliband, the former minister Miatta Fahnbulleh and the former Makerfield MP Josh Simons. The MPs Anneliese Midgley and Sally Jameson are closely involved.

Burnham’s team believe they can “lean in” to the Starmer government’s agenda as – unlike entering power straight after an election – a significant amount of work is already in motion. However, they said too much of this had been “underpowered” and they had to do a better job of showing the country it was delivering change.

Burnham would also reexamine Labour’s manifesto and look at areas where his government could be bolder, one source said. Another said the focus would be on four key areas: affordability, devolution, getting young people back to work, and ending the housing of asylum seekers in HMOs.

Burnham will begin to set out his policy platform next week with the first in a series of speeches to demonstrate a symbolic shift from Starmer’s government, starting with his plans for economic growth within the existing fiscal rules.

The new Makerfield MP has been keeping a lower profile in the 24 hours since he was welcomed by jubilant Labour MPs on his return to Westminster on Monday. He is expected to speak at a byelection campaign event in south London on Wednesday morning, but otherwise has been spending his time meeting groups of backbench MPs.

Burnham has been allocated an office in the Portcullis House complex of parliament, overlooking Big Ben, and on the same stretch of corridor as Haigh and Midgley. By coincidence Wes Streeting, who has backed him for leader, is several doors down.

Under the Labour party’s plans, if he was unopposed for the leadership, as expected, there would be a special conference to confirm the result on Friday 17 July. Burnham would then become prime minister on the same day.