Interviewed on BBC Radio 4's Today programme, Mr Johnson insisted staying in a customs union permanently would mean the United Kingdom being "essentially a colony'".
Mrs May said securing frictionless trade for goods was "one of our key negotiating objectives".
Writing her response to his letter of last Wednesday, Mrs May told the Labour leader: "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".
However she questioned his support for a customs union with the EU, saying it would prevent Britain from striking free trade deals with other countries around the world.
"Of course we always want to work with the opposition but the opposition has put forward some ideas that are not workable", he told reporters. Treasury Chief Secretary Liz Truss yesterday refused three times to say whether she would remain in the Cabinet if a customs union became official policy. She told Sky's Sophie Ridge on Sunday: "I absolutely do not think that should be our policy".
This is despite Mr Corbyn repeatedly saying there should be a general election if Mrs May can not get a deal through Parliament.
"It is very clear from the European Union that non-EU members do not have a say in EU trade policy so to pretend that you could do so is a unsafe delusion".
His comments were echoed by former Conservative chairman Grant Shapps who tweeted: "No point winning Labour MPs, by losing Tories!"
Simon Coveney has said what Ireland wants is certainty.
The former foreign secretary warned simply including the changes in a codicil to the Withdrawal Agreement would not be sufficient. "I think she feels, as I do, that there isn't actually as much dividing us from the Labour party as some people suggest".
May remains officially committed to getting the European Union to agree to significant changes to the Irish border backstop as a way of winning over the DUP and agitated Tory backbenchers who helped bring about the heavy defeat of her plan.
Mr Barclay will later travel to Brussels for talks over dinner with the EU's chief negotiator Michel Barnier.