Labour has called for the United Kingdom to join a new customs union after Brexit, saying it would leave the current one but negotiate a treaty afterwards that would "do the work of the customs union".
MPs are set to debate the Government's Brexit Trade Bill this week ahead of a key vote in May.
Peers voted on Wednesday in favour of an amendment to the flagship EU Withdrawal Bill which would seek to maintain the possibility that Britain might stay in a form of customs union.
The newspaper quoted one aide saying May would not be upset if Parliament forced her to stay in the customs framework by voting against her plans to leave it, though one person familiar with the prime minister's thinking disputed the account.
"I've heard many times the prime minister say that it is our policy to leave the Single Market and Customs Union".
She has said the United Kingdom would leave the EU's single market and customs union after Brexit in March 2019, noting the country may suffer new trade barriers as a result of her move.
The question of customs union membership goes to the heart of the kind of Brexit the United Kingdom will get and is likely to come to a head in a Commons vote as early as next month. It was in the Tory manifesto and Downing Street reiterated this stance on Monday.
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But it could give an indication of the opposition the prime minister might face in a crunch vote on the issue next month.
A customs union is when countries agree to apply the same taxes on imports to goods from outside the union.
Last week's defeat, the first of two on that day, increased pressure on May to reconsider her refusal to remain in a customs union with the EU.
BRITISH PRIME MINISTER Theresa May could have to row back on her earlier statement that the United Kingdom would leave the customs union after Brexit. His euroskeptic cabinet colleague, Liam Fox, the global trade secretary, is also a passionate advocate for leaving European Union common tariffs and commercial policies. Whatever one's view on the referendum result, it must be acknowledged that, outside the European Union, this country will enjoy a new degree of economic agility.
The Irish border is a key element of their case.
She said arrangements which are as "frictionless as possible" with the European Union and the ability to strike trade deals around the world were both achievable under the proposals set out by the Government.
"With Brexit looking increasingly like a risky threat to jobs and the economy in this country, the people have a right to make their voices heard with a People's Vote on the terms of the final Brexit deal".