May's Cabinet now has two proposals - the prime minister's preferred customs partnership model and the "max-fac" idea preferred by Brexiteers like Gove - which have both been rejected by the EU.
Environment Secretary Michael Gove, a leading pro-Brexit campaigner, said her customs partnership idea had "flaws", though he also acknowledged that neither plan was flawless.
The trio will develop the Government's alternative plan - known as "maximum facilitation" - which focuses on using technology to streamline customs checks.
Mrs May's official spokesman said that the issue was not discussed at Tuesday's regular meeting of Cabinet in 10 Downing Street, which Mr Johnson attended after returning from a visit to the US.
When asked on the BBC's Andrew Marr Show if he agreed with Johnson that the model is "crazy", Gove replied: "With the new customs partnership - Boris pointed out that because it's novel, because no model like this exists, there have to be significant questions marks over the deliverability of it on time".
The customs partnership had been Mrs May's preferred option.
The source, close to the European Parliament's Brexit taskforce, added that they feel a no-deal Brexit is "increasingly likely" with the British side leaning towards what Brussels regards as the least viable of the two options it has already rejected.
The other will look at the "customs partnership" - thought to be the Prime Minister's preferred option and supported by ministers who backed Remain - which would see Britain continue to collect tariffs on behalf of the EU. Regulations would also have to be aligned to a significant extent.
U.S. to help N Korea's economy if it abandons nuclear weapons
However, his comments made clear that the two sides remained far apart on the key issue of what they mean by denuclearisation. But he stressed that this would require a "robust verification" programme by the United States and other nations.
"This means there can be no hard border between Northern Ireland and Ireland, or between Northern Ireland and the rest of the U.K.", May wrote.
Speaking less than a week after fellow Brexiteer Boris Johnson called the proposal "crazy", the Environment Secretary claimed the policy would leave the United Kingdom acting as the EU's "tax collector".
May last week ordered her ministers to take responsibility for resolving the Brexit customs dilemma themselves, splitting her inner Brexit Cabinet into two working groups to iron out their differences. But some senior figures, including Mr Clark, have continued to push it.
It is the fourth poll in a row showing a four or five-point Tory lead over Labour.
"I trust the Prime Minister to do what she says she will do".
It came as former Labour home secretary Alan Johnson said the party could spend a further 18 years out of office and called for more moderate socialism.
The central role for Mr Davis reflects fears in government that he could quit if the customs partnership idea is pushed through.
Announcing the Commons schedule for the next fortnight, Andrea Leadsom, the leader of the house, gave no timings for the return of the European Union withdrawal bill from the Lords, or any news on three other key Brexit-connected bills.