In a sign the Government remained committed to its plan, he told BBC's Andrew Marr Show: "I think the only deal that we've got on the table, that's the Chequers deal".
Sarah Wollaston, chairwoman of the Commons Health Select Committee, said Mr Johnson was not "fit to lead the country" and indicated she would quit the party if he became prime minister.
The EU's chief Brexit negotiator, Michel Barnier, has sounded more hopeful of resolving the issue, offering to be flexible in his approach to agreeing the so-called backstop plan to avoid a hard border.
The environment secretary described rumours of a direct leadership challenge "whether it was by Boris (Johnson) or any other individual" as "loose talk" that would distract from the Brexit "mission". This is seen as critical by both sides, but talks have been deadlocked for months.
On Monday, Barnier said he thought reaching an agreement on the terms of the UK's departure was "realistic" within the next six to eight weeks.
"I think it's possible", Barnier said at a panel dubbed 'Bridging the Divide'. There is, so far, no full exit deal and it is uncertain whether May can push an agreement through the British parliament.
Barnier refused to comment on the divisions in the governing British Conservative Party over Brexit.
Once a deal is reached, it has to go to the United Kingdom parliament and the European Parliament for approval.
Mr Baker, a leading figure in the pro-Brexit Tory European Research Group, said the party's annual conference in Birmingham, starting on September 30, could prove a decisive moment as Mrs May is forced to acknowledge the scale of grassroots opposition to her proposals. According to two people familiar with the meeting, nobody spoke against the idea that she must be removed.
The trade union movement is preparing to throw its weight behind a public vote on the final Brexit deal because of fears the United Kingdom is going to "crash out" of the EU.
Theresa May should quit as Tory leader after Brexit day and hand over the torch to a "new generation leader" untainted by the European Union referendum, her former policy chief has said.
The ERG is unveiling its own alternative plan for tackling the Irish border issue on Wednesday.
Home Secretary Sajid Javid rebuked his former Cabinet colleague, saying "there are much better ways to articulate your differences". He urged Conservative lawmakers to get behind May, who he said was "doing a brilliant job" negotiating Brexit.
"Any diversion or any distraction from that mission means that our ability to ensure that the referendum mandate that we were given is delivered, is undermined". "We aren't expecting the European Union to change Barnier's guidelines, but we hope the leaders will tell them to interpret them in such a way as to make a deal possible", said one senior British official".