The deal reflects Switzerland's "mind the gap" strategy of ensuring seamless trade ties with Britain, regardless of whether London is able to strike and approve a formal exit agreement with Brussels by March 29, the date it is scheduled to leave.
Communities Secretary James Brokenshire said there might more non-binding votes on Brexit alternatives instead.
Should the parliament give May more time on Thursday, it would mark the second extension since her Brexit deal was defeated by MPs in January.
"We shouldn't be put in a position where the clock is run down and the prime minister says it's either my deal or even worse", Labour's Brexit pointman Keir Starmer told The Sunday Times.
Mrs May survived a confidence motion in her government tabled by Mr Corbyn, which could have led to an election, after her Brexit deal was overwhelmingly rejected by MPs last month.
May and her government have repeatedly said membership of a customs union would prevent it having an independent trade policy - something they have promoted as one of the main economic benefits of leaving the EU.
Shadow Brexit Secretary Sir Keir Starmer told the Sunday Times: 'We have got to put a hard stop into this running down the clock.
'We know that businesses are leaving the country, we know that businesses are making plans that will damage communities across the country and just this week we had a new chapter in the unfolding nightmare that the trade deals that the United Kingdom businesses enjoy through the European Union will not be ready in time for leaving'.
"We can't allow that to happen", Sir Keir said.
The UK has signed an agreement with Switzerland ensuring the trading relationship between the two nations will continue after Brexit, the British government said. 'There needs to be a day when Parliament says that's it, enough is enough'.
Mrs May described discussions with European Union leaders in Brussels as "robust but constructive" and insisted she was determined to "negotiate hard" over the coming days to secure legally-binding changes to the Agreement which will render it acceptable to Parliament.
The impasse has left the world's fifth largest economy facing an uncertain future, rattling financial markets and businesses about the prospect of a disorderly exit from the bloc that could damage the economy.
But the Irish government and the European Union have repeatedly rejected calls for changes.