Other lawmakers, and many businesses, want Britain to maintain customs alignment with the European Union to avoid barriers with the U.K.'s biggest trading partner.
On the eve of the Brexit cabinet meeting, Brexiteers urged Mrs May to abandon the partnership option, presenting a 30-page dossier claiming it would make meaningful trade deals "impossible" to forge and render the UK's International Trade Department "obsolete".
Every living former Cabinet Secretary voted against the government to keep Britain in a customs union with the European Union in a recent Lords vote, with Lord Robin Butler - who served under former prime ministers Margaret Thatcher, John Major, and Tony Blair - wailing that the thought of leaving the bloc "strikes a dagger to my soul". The EU says agreement on whether the United Kingdom stays in any form of customs arrangement has to have been reached by next month.
The government's decision on its preferred customs option was postponed this week after senior ministers failed to reach agreement over a new model.
A number of Cabinet ministers spoke out against the Prime Minister's hybrid "customs partnership" model which would see Britain collect tariffs on behalf of the European Union for goods destined for the block, with firms potentially claiming back a rebate if products remained in the United Kingdom on a lower-tariff regime.
Business secretary Greg Clark is said to have been "close to tears" as he warned of the potential impact on jobs if the Government abandoned the customs partnership idea.
Some of her Tory peers have also pushed for Mrs May to abandon her red lines and keep the United Kingdom in the single market as part of a desperate Remoaner effort.
Britain rejects one suggestion from the EU - to keep Northern Ireland inside the bloc's customs union.
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However, Labour peers have reportedly been told to abstain.
He said: "What we have said very clearly, and I think actually quite a lot of the Conservatives are going to follow us in this, we remain within the customs union during the transition period, we want to negotiate a customs union, that will solve the Northern Ireland border problem, which I think is intractable".
Labour's Lord Alli, one of the signatories to the amendment, accused the party leadership of being "paralysed by indecision".
What is a customs union?
The UK, which put forward two alternative proposals past year, has yet to confirm its favoured model.
A customs partnership - This would remove the need for new customs checks at the border.
This would potentially allow businesses to claim back a rebate if products remained in the United Kingdom on a lower-tariff regime. Poor developing countries are not high on the government's list, which seems to start with Australia, Canada, New Zealand and the US (Has anyone noticed what differentiates those countries from an African or Caribbean country?).