The Prime Minister struck a more emollient tone than recent weeks by calling called for further talks with the Opposition on how to secure a Brexit deal that could command cross-party support.
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.
As the Prime Minister prepares to update MPs on the latest developments in the Brexit negotiations on Tuesday afternoon - a day earlier than expected - Andrea Leadsom dismissed the prospect of Mrs May adopting Jeremy Corbyn's "world view".
Labour wants to force the government to bring back a "meaningful vote" on Theresa May's Brexit deal by the end of this month.
Her response comes ahead of another week of Brexit debate and voting in Westminster.
As the clock ticks down to Britain's scheduled exit on March 29, May is trying to persuade the European Union to change a deal that was agreed between London and Brussels late past year but overwhelmingly rejected by parliament in January.
"I absolutely do not think that should be our policy", she told Sky News on Sunday.More news: French 'yellow vest' protester loses fingers in violent clashes
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Last week, Labour leader Jeremy Corbyn set out the conditions under which he would instruct his party to support an exit deal in parliament.
"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". One, which called for permanent membership of the customs union, would have guaranteed a very soft Brexit.
Mr Corbyn's demand is for a customs union that gives the United Kingdom a say on future trade deals the EU might strike - something Brussels appears unlikely to accept.
At the meeting, Department for Trade officials were probed on the pledge by Mr Fox that he would sign free trade agreements to replicate the EU's 40 existing deals with countries around the world.
He added: "I am concerned that with 46 days to go the Prime Minister is simply running down the clock, mindful that the next EU Summit is March 21 and if she's is trying to run the clock down Parliament has to step in with a hard stop to say we are not going to accept that".
Mr Johnson said Mrs May's priority should be to secure a time limit on the backstop - meant to prevent the return of a hard border on the island of Ireland - before the next election scheduled for June 2022. 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.
May's Brexit secretary, Stephen Barclay, traveled to Strasbourg on Monday to meet the EU's chief negotiator, Michel Barnier, and talk about the Irish backstop issue, which aims to avoid a hard border between Northern Ireland and the Irish Republic by binding the United Kingdom to European Union rules until a free-trade deal is inked, something critics fear could indefinitely trap the United Kingdom in a watered-down version of European Union membership.