He is also lining up UK interior minister Shabana Mahmood to be his chancellor, the newspaper reported, adding that talks between Shabana's team and Burnham had been going on for a while.
Former United Kingdom deputy prime minister Angela Rayner, who said earlier this month that she has been cleared of deliberate wrongdoing over her tax affairs, is expected to be brought back into the cabinet if Burnham succeeds in his leadership bid, the report said.
Reuters could not immediately verify the report.
Burnham, Rayner and Mahmood did not immediately respond to requests for comment.
Burnham, seen as a challenger to embattled Prime Minister Keir Starmer, would need to win the Makerfield seat near Manchester before launching any formal leadership challenge.
A by-election for a seat in the UK parliament will take place on June 18, the local council in northern England said in a notice earlier this month.
Starmer's Labour Party performed poorly in May local elections, triggering calls from a significant bloc of his MPs for him to quit.
Many of those are looking to Burnham to revive the party's fortunes.
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