He had previously said that despite seeing off the motion of no confidence by 200 votes to 117 she should resign "as soon as the Queen has a moment in her diary to see her".
Labour's motion was not put forward under legislation that determines when the United Kingdom holds general elections, and would not have collapsed the government.
Ms Rudd said this morning that Parliament should be allowed to express its will and "nothing should be off the table".
Barwell apparently thinks it's the only way May can end her Brexit nightmare.
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. "Fair enough, we're now going for it".
May accused Blair of insulting voters and trying to undermine her government by meeting officials in Brussels. "When there are only three months left to avert disaster, that's really not excellent tactics", she added. "Parliament has a democratic duty to deliver what the British people voted for".
"The prime minister has obdurately refused to ensure a vote took place on the date she agreed, she refuses to allow a vote to take place this week and is now, I assume, thinking the vote will be on the 14 January, nearly a month away".
She added: "We can not, as he would, abdicate responsibility for this decision".
In the Commons, the Prime Minister will all but rule out holding a second vote while she remains in charge.
Updating the Commons on a turbulent summit in Brussels last week, Mrs May told MPs that "some of the resulting exchanges at this Council were robust".
Theresa May announces when MPs will vote on Brexit deal
But with the European Union unlikely to offer concessions that would win over lawmakers, more politicians are backing a second referendum - something some of her ministers say could be avoided if the government tested Brexit scenarios in parliamentary votes.
May said parliament would debate the deal in January, before a vote in the week beginning Jan. 14 - more than a month after an original December 11 vote which May cancelled after admitting she faced a significant defeat.
Speaking last week to KMTV, Mr Green said: "If the deal falls I think we just need the full range of options put before Parliament, including Norway, Canada, a second referendum, a "No Deal" - all of which I think are worse than the current deal".
"The prime minister has cynically run down the clock and tried to manoeuvre parliament into a choice between two unacceptable outcomes: her deal or no deal", he added.
The SNP's Westminster leader Ian Blackford said it was "time to call time on this government" and taunted Mrs May: "Where is the leadership?"
But May and her ministers have repeatedly ruled out a new ballot, saying it would deepen divisions over Britain's biggest decision since the Second World War and betray voters who narrowly backed leaving the European Union at a 2016 referendum.
The longer it takes, voices urging a change of tack are getting louder and the pressure on the main opposition Labour Party to move against the government is rising.
Cabinet ministers are openly discussing how to respond to Theresa May's Brexit deal being rejected by MPs, with the possibility of the Commons being asked to choose from a menu of alternatives in order to break the deadlock.
Meanwhile, PM also said that MPs should not be helping Corbyn trigger a general election.
"The deal that is on the table is the best and the only deal possible - we will not reopen it, it will not be renegotiated", he said.
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