Fox had previously said Britain would be able to roll over all of the EU's current trade deals, but so far he has only concluded agreements with Chile, the Faroe Islands and Switzerland.
I know that sounds nebulous, a word you might choose to use.
In her letter, Mrs May said she wanted discussions between Tory and Labour teams to start considering "alternative arrangements" to the backstop contained within the withdrawal agreement.
He said the ideas put forward by Corbyn were "not workable", because it was impossible to subscribe to his proposal to have a customs union with the EU while also influencing EU policy.
Labour's Brexit policy chief Keir Starmer told the Sunday Times newspaper that his party would seek to use the debate in parliament this week to prevent May from waiting until the last minute to come back with a deal, and compel her to present a fresh accord for lawmakers to consider before February 26.More news: Offset Releases Footage of Cardi B Giving Birth in New Album Trailer
The existing Political Declaration - the part of the Brexit deal setting out the goals for the future UK-EU relationship - "explicitly provides for the benefits of a customs union - no tariffs, fees, charges or quantitative restrictions across all sectors and no checks on rules of origin", said the PM. He also said that a "free trade deal with the European Union should be the easiest in human history".
It comes as the government promised MPs more Brexit votes before the end of February amid claims Mrs May is trying to run down the clock.
But Mr Corbyn's demand is for a customs union that gives the United Kingdom a say on future trade deals the EU might strike and is something Brussels appears unlikely to accept.
She also questioned whether the call for completely "frictionless" trade would mean reneging on Labour's commitment to end free movement by requiring single market membership.More news: Katy Perry Pulls ‘Blackface’ Shoes From Retailers
The government announced on Sunday that parliament might not be given the chance to vote on its Brexit deal until March.
May is already due to update parliament on her progress towards a deal on Wednesday and then on Thursday to give parliament a chance to express their opinion.
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.
The comments reflect concern among Brexiteers that Mrs May is preparing to concede too much ground to Labour in an attempt to win cross-party backing for her deal with Brussels.
The prime minister said in her letter: "I have always been clear that Brexit should not be at the expense of workers' rights or environmental protections".More news: 2019 iPhones reportedly dash our USB-C hopes and dreams
The move is aimed at postponing a rebellion by ministers who want to remove the possibility a no-deal Brexit.