Wales minister and whip Nigel Adams said the Prime Minister had made a "grave error" by reaching out to the Corbyn.
This afternoon Brexit junior minister Chris Heaton-Harris also quit, saying Britain should have fulfilled the result of the 2016 referendum by leaving the EU on March 29.
Britain risks crashes out of the EU without a deal – with the Commons voting down everything presented to them amid bitter divisions.
Mrs May announced her dramatic move after an eight-hour marathon session of Cabinet.
In his resignation letter, Adams said: “I and many others agreed with your previous position that no deal is better than a bad deal.
“It now seems that you and your cabinet have decided that a deal – cooked up with a Marxist who has never once in his political life, put British interests first – is better than no deal.”
“A deal cooked up with a Marxist”
Heaton-Harris said May's clear desire not to leave with no deal made his work in the Brexit department "irrelevant" forcing his hand.
Adams had become the 35th resignation to hit Mays government over the past year as the Tories tear themselves to pieces over Brexit.
High profile quitters include Boris Johnson, David Davis, Dominic Raab and Esther McVey.
The MP said that the Government faced two "great challenges" of delivering "the Brexit the people voted for" and preventing a Corbyn premiership.
And he added: "Sadly, I fear that we are now at risk of simultaneously failing in both."
ANOTHER ONE BITES THE DUST: Nigel Adams just became the 35th minister to quit the Government (Pic: PA)
May and Corbyn are expected to meet today in bid to come to some kind of agreement over Britains exit from the EU.
Corbyn said he would be "very happy" to meet the Prime Minister in a bid to offer "certainty and security" for the UK.
Tory Brexiteers have reacted with anger and accused May of handing control of the negotiations to Labour.
Jacob Rees-Mogg described the offer as "deeply unsatisfactory" and accused Mrs May of planning to collaborate with "a known Marxist".
Former foreign secretary Boris Johnson said: "It is very disappointing that the Cabinet has decided to entrust the final handling of Brexit to Jeremy Corbyn and the Labour Party."
DOWNING STREET: Theresa May's days in Number 10 could be numbered amid Brexit (Pic: GETTY) Related Articles
HOLDING THE CARDS: Jeremy Corbyn is being courted by Theresa May to solve Brexit (Pic: GETTY)
May is fighting for her future as Prime Minister amid reports she will resign as soon as negotiations are done.
Brexiteers have been mounting pressure for May to go as she has failed three times to get her Withdrawal Agreement through the Commons.
Elsewhere, EU chiefs will hold the first of a series of briefings on Brexit preparedness ramping up arrangements for “no deal”.
European Parliament Brexit coordinator Guy Verhofstadt, who had said he thought a no-deal Brexit was "nearly inevitable", welcomed Mrs May's offer of talks with Mr Corbyn.
FURIOUS: Jacob Rees-Mogg and other hardliners have accused Theresa May of 'courting a MarxisRead More – Source