Scottish Secretary David Mundell said his party was “certainly willing” to discuss that in talks with Jeremy Corbyn’s party - contrary to the Prime Minister’s previous stance that leaving the European Union would mean Britain coming out of both the customs unions and the single market.
He spoke after Mrs May managed to secure a second extension for the UK’s Article 50 process, which could see the UK stay in the EU until the end of October.
Scotland’s First Minister Nicola Sturgeon - who is a leading advocate for a People’s Vote - has insisted the UK must “not waste this time”.
She tweeted her “relief that thanks to the patience of the EU” the UK would not be “crashing out” under a no-deal Brexit on Friday.
And she stressed that “allowing people to decide if they still want to leave is now imperative”.
Mr Mundell refused to say if the prospect of a second Brexit referendum had been discussed in talks between the Government and Labour.
But he told BBC Radio Scotland’s Good Morning Scotland programme: “The Government is certainly willing to discuss a customs union, but a customs union would require to command a majority of support in Parliament.
“A customs union has been put forward previously in Parliament and hasn’t commanded a majority, partly because we’ve had the usually politicking, the SNP who say they support a customs union then didn’t vote for it.
“So nothing that goes forward will actually be successful unless we can command majority support for it in Parliament.”
He insisted Mrs May could lead the Conservatives in a European election campaign, if Britain is required to take part in the May ballot.
But he also argued that the Prime Minister could still get her Withdrawal Agreement through the House of Commons in time to prevent this
Mr Mundell said: “Mrs May wants to deliver Brexit by June 30, indeed she wants to deliver Brexit by May 22 so we don’t have to have the European elections.”
He added that there was “still an opportunity to do that”, as the discussions with Mr Corbyn’s party “seem to be serious”.
If the Government could then “get some form of agreement with the Labour Party, then it would be possible to ratify the Withdrawal Agreement by May 22 and leave by then”, he said.
If the UK has to take part in European elections, he argued, Mrs May would “certainly” lead the Conservative Party into any campaign “because of the timescale of those elections”.