May accused Blair of insulting voters and trying to undermine her government with calls for a second referendum to break the political deadlock over the divorce deal she struck with the EU.
There were reports in the Sunday newspapers that two of Mrs May's key allies were planning for another referendum, in the event that her deal can not get MPs' backing.
And she added that there were too many people trying to "subvert" Britain's withdrawal from the European Union for their "own political interests" rather than those of the nation.
But the European Union leaders rebuffed British pressure to put a fixed end date on the border guarantee, and refused to re-negotiate the 585-page legal text settling issues of the divorce.
On Thursday about 10 Labour MPs met David Lidington - who is Mrs May's de facto second-in-command - to argue for another public vote.
And David Lidington also took to Twitter to post an excerpt of a speech he made this week in the House of Commons in which he said he had come to terms with being on the losing side of the 2016 referendum.
Many senior Labour figures are deeply uneasy about endorsing another referendum. "We've hadthe people's vote, we've had the referendum and now we've got to get on with implementing it".
"Happy to confirm I am not planning a 2nd referendum with political opponents (or anyone else, to anticipate the next question)". "Where does that end up?"
British PM delays Brexit deal vote but European Union warns no renegotiation
Parliament has until January 21 to vote on the deal, a little more than two months before Britain's March 29 departure date. A British Conservative Party official says Prime Minister Theresa May will face a no-confidence vote from party lawmakers.
What has the potential to swing the dial is that members of the cabinet are now publicly advocating free votes in parliament, to test the levels of support for different options - including a second referendum.
"The sensible thing is now to allow parliament to vote on each of the forms of Brexit canvassed, including the prime minister's deal", he said in a statement.
Weeks ago only a marginal issue, now dozens of Labour MPs are backing another vote, as well as the SNP, Liberal Democrats, and smaller parties.
But Labour frontbencher Andrew Gwynne told the BBC: "We can't move to the next stage until Parliament has decided whether or not to back the prime minister's deal".
"What is irresponsible, however, is to try to steamroll MPs into accepting a deal they genuinely think is a bad one with the threat that if they do not fall into line, the Government will have the country crash out without a deal", he said.
Nile Gardiner, an aide to former prime minister Margaret Thatcher now at the Heritage Foundation, a conservative U.S. think tank, said Brexit was a natural philosophical fit for Trump.
With less than four months before Britain is due to leave in March, Brexit, the biggest shift in trade and foreign policy for more than 40 years, is proving anything but smooth, complicated by the deep divisions in Parliament and across the country.
Other cabinet ministers are also reported to favour a scenario of asking MPs to vote on options, which could include a no-deal Brexit, a second referendum, and a "Norway option" to keep closer economic ties with the EU.
He urged politicians to back the PM's plan, describing it as "balanced" and the "best of both worlds".