The prime minister queried his call for the United Kingdom to stay in a customs union with the EU - but welcomed more talks with Labour on a Brexit agreement.
Mrs Leadsom, an ardent Leaver, said she would stay in the Cabinet to help Mrs May deliver Brexit and denied that the PM was softening her stance over a customs union in a letter to Mr Corbyn.
Trade secretary Liam Fox said the government wanted to work with Labour, but it has "put forward ideas that are not workable".
That might mean that she's back before the end of this month with something concrete to vote on.
She'll say that if she hasn't brought them new deal by February 27, there will then be another opportunity to vote, Communities Secretary James Brokenshire confirmed in an interview.
Boris Johnson said a permanent customs union would turn Britain "essentially into a colony of the EU".
Mrs May said: "It is good to see that we agree that the United Kingdom should leave the European Union with a deal and that the urgent task at hand is to find a deal that honours our commitments to the people of Northern Ireland, can command support in Parliament and can be negotiated with the EU - not to seek an election or second referendum".
Corbyn had earlier laid out five Labour Party conditions for supporting any revised Brexit deal.
"We would have to be able to get out by a certain time and we would have to be able to get out of our own volition".
She talked up hope of a breakthrough with Brussels, insisting: "I believe we can reach a deal that this House can support".
They have also signed a deal to preserve the rights of some 43,000 British nationals living in Switzerland and the 35,000 Swiss residing in Britain after Brexit.
The government has signed a trade deal with Switzerland, heralded as the most significant such agreement in the run-up to Brexit.
For remaining European Union members, Mr Bettel says the ongoing uncertainty in London "puts us in a position where we don't know what is likely to happen tomorrow".
"I am not clear why you believe it would be preferable to seek a say in future European Union trade deals rather than the ability to strike our own deal?"
Mrs May said securing frictionless trade for goods was "one of our key negotiating objectives".
"I think it must be pretty obvious that if you are going to have a time limit to the backstop, and I think that would be a very good thing, it's got to fall before the next election". Although lawmakers asked for the backstop to be removed, May has said since that the backstop will remain, and that Parliament merely asked for it to be altered.
Mr Barclay was said to have held "constructive" talks on Monday night with the EU's chief negotiator Michel Barnier in Brussels.
In the meantime, Brexit Secretary Stephen Barclay has been meeting members of the Alternative Arrangements Working Group of Tories drawn from all sides of the party who have been seeking a compromise solution to avoid the need for the controversial backstop.