In recent debates we have set out new priorities, such as stemming illegal migration, improving European security and defence as well as investing more in the Erasmus programme.
Officials said leaders were mostly opposed to what they see as a power grab by MEPs at the expense of national sovereignty.
Labour leader Jeremy Corbyn is set to unveil a competing vision of how Brexit should work in what is being billed as a significant address on Monday. MPs are expected to vote on the bill within weeks, but as May only has a working majority of 13, the likelihood of the bill being passed has tipped in Soubry's favor - and it would significantly undermine the PM's authority... again.
Brexit-supporting Conservative MP Michael Fabricant described it as a "total betrayal" of Leave voters.
Speaking to the BBC, Health and Social Care Secretary Jeremy Hunt rejected the customs union calls and said it was possible to have "frictionless" trade with the continent without one.
The threat came following an eight hour gathering of senior Cabinet members, who met to find a united front on European Union withdrawal.
Thursday also saw the government's Brexit sub-committee hold talks for eight hours at Chequers on how closely the European Union and United Kingdom should be aligned after exit day in March 2019.
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"When we try to convince Prime Minister May to come forward with her vision of the future, I don't think the three-basket approach is what we have in mind", a senior European Union diplomat added.
Mr Hunt - who was not at Chequers - said he had been told it was "positive and constructive".
"But that must always be on a voluntary basis", he told Sky News.
"We will as a sovereign power have the right to choose to diverge and what we won't be doing is accepting changes in rules because the European Union unilaterally chooses to make those changes", Mr Hunt told the BBC.
He will also push ahead with that even if there is no deal on a post-Brexit transition period, now mired in negotiations about how long it will last, how closely Britain must stick to EU laws, and the rights of European citizens.
Such a move would represent a major climb-down by Mrs May after she insisted those arriving after March 2019 should not have the same rights as people who came before.
The full cabinet will discuss the plans at their regular weekly meeting on Tuesday.