In a statement, he stated: "This towns fund smacks of desperation from a government reduced to bribing Members of Parliament to vote for their damaging flagship Brexit legislation".
As Brexit goes down to the line, investors are watching to see if May can win over enough MPs to her deal: if she cannot, then the exit date is nearly certain to be delayed by MPs eager to avoid a potentially disorderly no-deal exit.
Theresa May announced that there is £1.6 billion in regional development funds.
But the Daily Telegraph newspaper said Cox had abandoned attempts to secure a hard time-limit or unilateral exit mechanism. EURACTIV has travelled to Northern Ireland where Brexit is more than a political or economic issue - it is a challenge for peace.
The DUP and the Tory European Research Group (ERG) have made it clear they will not support the PM's withdrawal agreement in crunch Commons votes next week without such legally binding measures.
A source close to a Cabinet minister told the liberal-left newspaper that there was "no chance" of the agreement passing next week, with Brexiteers warning May not to try to ambush them with a last-minute deal with the European Union on the Irish backstop - a sticking point for the DUP and Eurosceptics - and expect them to support the Agreement without giving the document proper analysis.
The prime minister's spokesman dismissed accusations of a "Brexit bung" on Monday and said the money was "absolutely not" dependent on votes.
Mrs May has promised MPs a "meaningful vote" on her deal by March 12.More news: R. Kelly going to jail after failing to pay child support
Prime Minister Theresa May said: "For too long in our country prosperity has been unfairly spread".
"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".
"These towns have a glorious heritage, huge potential and, with the right help, a bright future ahead of them", she added.
Shadow chancellor John McDonnell branded the initiative a "desperate bribe".
Almost three quarters of Labour voters across the East Midlands back Jeremy Corbyn's decision to now support a People's Vote on Brexit, according to a YouGov opinion poll.
Most of the initial one billion pounds will go to communities in the English north and Midlands, where many Labour-held parliamentary constituencies returned a majority for Brexit in the 2016 referendum.
The two sides are at a "critical point" in these negotiations, a spokeswoman for May said on Tuesday. Parliament has been clear that for this to happen, we require legally-binding changes which mean that the United Kingdom can't be trapped in the backstop indefinitely. European Union leaders insist that the legally binding Brexit withdrawal agreement can't be reopened, and talks are focusing on drafting an addendum or other additional words.
"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".