In her letter, May questioned Corbyn's call for the United Kingdom to remain in a customs union with Brussels but said she wanted further talks with the Opposition on a Brexit deal.
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.
Speaking to BBC radio Monday morning, Johnson said: "The argument is now about how to get out of the backstop". I think you would need to have a time limit.
With a vote due on February 14, May will ask Parliament to reaffirm its desire to remove the contentious Irish backstop clause from the Withdrawal Agreement, according to an official, who asked not to be identified.
She insisted her deal already met numerous conditions he had set.
The intervention came as the PM (pictured left returning to No10 today) moved to quell Tory fears that she is about to cave into Jeremy Corbyn's (pictured right) demand for a permanent customs union with the EU.
"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 deals?" she said.
She also questioned whether the call for "frictionless" trade would mean reneging on Labour's commitment to end free movement. "Mrs May also said the Tories were "prepared to commit" to new laws to protect workers" rights after Brexit - a key demand of Labour and the unions.
The prime minister struck a conciliatory tone in her response overnight and said she looked forward to the two parties meeting again "as soon as possible" to discuss ways forward on Brexit.
Mr Corbyn has repeatedly said there should be an election if Mrs May can not get a deal through Parliament and he has faced concerted pressure from some in his party to push for a second public vote.
May wants to win over lawmakers in her Conservative Party with changes relating to the Northern Irish border, but the EU has refused to reopen that part of the deal and instead wants May to pursue a compromise with the main opposition Labour Party by agreeing closer UK-EU ties.
Labour will use a vote expected on Thursday to attempt to force the Prime Minister to bring the deal back for a showdown by February 26 to prevent her "running down the clock" before Brexit.
Challenged on whether there would be a vote on the deal this month, rather than on another series of amendments, he said: "If the meaningful vote has not happened, so in other words things have not concluded, then Parliament would have that further opportunity by no later than 27 February".