May has struggled to convince the European Union that she can get the deal through a deeply divided Parliament in London, where lawmakers are increasingly flexing their muscles to try to influence the U.K.'s departure from the bloc.
There is no direct allocation for Scotland announced in the funding plan.
In a further sign that former opponents of May's deal might now back a revised version, Graham Brady, a senior Conservative lawmaker, said he would support it with legally binding assurances on the backstop.
The attorney general is trying to secure an enhanced "arbitration mechanism" that allows Britain or the European Union to provide formal notice that the backstop should come to an end, the Telegraph said. "The negotiations are at a critical and sensitive point".
She has offered legislators the chance to seek to prevent a no-deal departure and to delay Brexit if Parliament rejects the deal in a vote she has promised to hold by March 12.
"For too long in our country prosperity has been unfairly spread", said May.
"Communities across the country voted for Brexit as an expression of their desire to see change, that must be a change for the better, with more opportunity and greater control".
She said: "After decades of neglect a one-off payment created to help the Prime Minister ahead of a key Brexit vote will fail, and it will confirm to people in our towns that the government is not serious in its commitment to our communities".
The Labour Party's finance spokesman, John McDonnell, said the fund was "Brexit bribery".
The threat of Brexit, especially if there's no deal, could hurt the British economy.
If it's rejected again, MPs will then vote on whether or not to reject a no-deal Brexit.
Another £600m will be available through a bidding process to communities in any part of the country.
One EU diplomat said Mr Cox's proposal would give the United Kingdom the right to say it had acted in good faith but that all efforts to replace the backstop had come to nought and therefore it could walk away.
The group wants a legally binding, treaty level arrangement, the language must not just simply reiterate the temporary nature of the backstop, and there "needs to be a clear and unconditional route out" of the arrangement.
While the British government has sought concessions, European Union leaders have repeatedly ruled out the prospect of substantial renegotiation of the withdrawal agreement.
"And which makes clear that it can not be indefinite, and it can not be a trap both for the United Kingdom generally and for Northern Ireland in particular".
British Prime Minister Theresa May unveiled £1.6 billion (US$2.12 billion, 1.87 billion euros) in regional development funds Monday in what opposition MPs dismissed as a "desperate bribe" to persuade them to back her Brexit deal.