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.
The prime minister's formal response to Jeremy Corbyn's proposal, in a letter to the Labour leader, stressed her objections to keeping the United Kingdom in some form of customs union, saying this would prevent the United Kingdom making its own trade deals.
She did not reject any of his conditions outright in her reply.
This is despite Mr Corbyn repeatedly saying there should be a general election if Mrs May can not get a deal through Parliament.
Senior Tories have warned Labour's plans for a customs union with the EU are a "dangerous delusion", after Theresa May offered fresh Brexit talks with Jeremy Corbyn.
"The fundamental negotiating challenge here is the EU's position that completely frictionless trade is only possible if the United Kingdom stays in the single market", she said.
She wrote: "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 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".
Downing Street meanwhile said Mrs May will make a Commons statement on the latest developments in the Brexit negotiations on Tuesday - a day earlier than expected.
She added: "The fundamental negotiating challenge here is the EU's position that completely frictionless trade is only possible if the United Kingdom stays in the single market".
Nevertheless, the Prime Minister told Mr Corbyn she looked forward to their teams meeting "as soon as possible".
In a letter to Tory Party chair Brandon Lewis, the activists said any attempt to work with Labour to secure a deal or renege on manifesto commitments would be "catastrophic" for the party and lead to "severe electoral defeats" in the forthcoming May local elections.