The proposal comes ahead of any second vote on the prime minister's deal and after Ms May secured parliamentary backing to seek alternatives to the Irish backstop.
A Downing Street source confirmed to The Independent the funds for Leave areas was a "possibility" but made clear it had not yet been agreed.
John Mann - the Labour MP for Bassetlaw - said: "Show us the money. A fund of sufficient size to transform our communities. Our areas voted Leave and it is time that we had the investment we need."
Mr Mann was one of 14 Labour MPs to vote against an amendment by his colleague Yvette Cooper on Tuesday which would have paved the way to delay Britain's exit from the EU by giving MPs the power to request an Article 50 extension.
Downing Street hope they can win the support of Labour MPs when Ms May returns to the Commons with a second "meaningful vote" on her deal next month as it is expected several Conservative MPs will still remain opposed to her plans.
In regards to the cash injection, a government source also told The Times: "There's a willingness to look again at coalfield communities and make good the promises that former Labour governments failed to deliver.
"It's about allowing Labour MPs representing Brexit communities to show that they have extracted something in return for their vote. And, frankly, it's not an unreasonable ask."
In a warning to colleagues tempted to support the prime minister's deal in the coming weeks, however, the Labour MP David Lammy said: "More fool them. Socialists my arse. Cowards and facilitators. History will be brutal."
It comes as Ms May prepares to spend the coming weeks attempting to renegotiate the withdrawal agreement and the Irish backstop, despite the EU repeatedly rejecting such demands from the UK government.
On Wednesday, Jeremy Corbyn held talks with Ms May on Wednesday in which he "set out the Labour case for a comprehensive customs union with the European Union".
But speaking to reporters in his parliamentary office after the meeting, he warned that he had suspicions about the government’s motives in holding the meetings with opposition parliamentarians.
The Labour leader said: “The whole process looks like it’s running down the clock by saying, well, it’s either the problems and the difficulties of no deal or support a deal that’s already been rejected by the House of Commons.
“I’m suspicious that there is a programme of running down the clock here.”
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