The former Prime Minister, who only stood down last month, considers his return a matter of public interest, reports The Times.
A new prime minister will be chosen next week after Liz Truss announced her resignation.
After insisting less than 24 hours earlier she is a “fighter, not a quitter”, the Conservative party leader stood at a podium outside 10 Downing Street on Thursday to say her brief time in the top job will come to an end.
There are now growing calls for an early general election.
Sir Keir Starmer said: “The risk at the moment is continuing with this chaos, not having a stable Labour government.
“So that’s why there needs to be a general election.
“We can’t just allow the Tory party to keep putting up the next candidate in the middle of this chaos.
“There is a choice, there’s a Labour Party that’s capable of stabilising the economy, has a clear plan and the public are entitled to choose between that stable Labour government and this utter chaos.
Liz Truss apologises for mistakes made as Prime Minister
“We are ready to form a government, to stabilise the economy and implement a real plan for growth, for living standards, to help people through a cost-of-living crisis.
“And that’s the choice now, a stable Labour government or this utter chaos from the Conservatives.”