Both include ministers from the pro- and anti-Brexit factions, but two of the most hardline on each side - Foreign Secretary Boris Johnson, responsible for the "crazy" barb, and Chancellor of the Exchequer Philip Hammond - aren't included.
Gove, one of the leading Leave campaigners ahead of the 2016 referendum, said the result of that vote showed the British people want the United Kingdom to be free from the EU customs union.
She said: 'You can trust me to deliver.
"Of course, the details are incredibly complex and, as in any negotiation, there will have to be compromises".
Looking at the customs partnership proposal will be two Brexiteers - International Trade Secretary Liam Fox and Environment Secretary Michael Gove - and Remainer Cabinet Office minister David Lidington.
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The resignation of a key ally left Mrs May outnumbered in her inner "war cabinet" over her proposal for a close customs relationship with the bloc, while the Sunday Telegraph reported that at least 12 ministers - of 28 who sit in Cabinet with Mrs May - are set to oppose her plan.
As well as trade, the decision is critical to how Mrs May plans to meet her pledge to avoid a hard border in Ireland after Brexit.
"Any agreements must create as little friction as possible for trade", she said, adding: "We must not constrain our ability to negotiate trade agreements with other countries around the world by being bound into a customs union".
May's preferred option, the customs partnership, was reportedly rejected at a meeting of her senior ministers last week, while Brussels has also condemned it as "magical thinking". The EU has expressed doubts about whether either option would work.
The Tory MP, who is also chair of the Treasury Select Committee, also criticised the other proposal being put forward, and said: "It seems to me that's what's called the maximum facilitation - which seems to rely on future technology not yet invented - would absolutely basically be a deal in name only because it doesn't talk about an enduring relationship with the European Union, which I think is what the Prime Minister said she wanted to create, and it causes enormous problems on the island of Ireland".
A decision is not expected for at least another week, but the clock is ticking ahead of a crucial European Union summit in June.