A cabinet minister said leader of the pro-Brexit European Reform Group Jacob Rees-Mogg also supports the plan, while other Brexiteers said it could be worth exploring in more detail.
Calling for a second referendum to be held on Brexit, Mr Johnson denounced the choice between Mrs May's deal plans or a no-deal scenario as a "failure of British statecraft on a scale unseen since the Suez crisis" that had left Britain facing "vassalage" or "chaos".
On Friday the DUP, whose support Theresa May relies on for votes in the Commons, said they can not support any deal which included the possibility that Northern Ireland would be treated differently from the rest of the UK.
Johnson, the younger brother of Boris Johnson, stepped down as a transport minister Friday and called for a second Brexit referendum.More news: Stephen Hawking's wheelchair and thesis sold at auction
Asked if his brother Boris Johnson had lied to voters during the referendum about Brexit, the former transport minister said: "In the campaign there were undoubtedly promises made that have shown to be undeliverable".
He said the public should be asked to confirm their decision to leave the European Union and, if they choose to do that, whether to accept Mrs May's plan or leave without a deal.
While the majority of his constituents voted to leave the European Union, he rejected the idea that a vote on "what an idealised Brexit might offer" versus what was being delivered in reality, matched up to what many had voted for, describing the current state of things as an affront to democracy.
"We can stop this Tory Brexit but, ultimately, it should be for the people to decide", he said.
Another brother, Leo, 50, is not in public life but, as a remainer, he tweeted: "Respect to you Jo my bro".More news: Kim Kardashian shares images while fleeing massive California wildfire
Johnson, backed the Remain side during the referendum said it would be "a travesty" not to have a second referendum.
Instead, Johnson said it would be proper to give the British public a chance to vote again because of the way the Brexit negotiations have evolved.
She has consistently rejected the idea of another nationwide vote on Brexit, insisting her obligation is to make good on the will of the people as expressed in 2016. Meanwhile, a spokesman for May's office says there will not be a second referendum on Brexit under any circumstances.
Like his older brother, Mr Johnson enjoyed a distinguished career in journalism before entering politics, working for the Financial Times as, among other roles, South East Asia bureau chief and Paris correspondent.More news: WWI: Macron and May pay tribute at Somme