Though he was reported to have said last week that his focus was on forcing Mrs May to ditch her plan, speculation about a leadership challenge isn't going away.
The United Kingdom and European Union may still be far apart, but there's movement - enough for currency traders to push the pound higher.
May remains confident of striking an acceptable deal with Brussels.
International Monetary Fund (IMF) managing director Christine Lagarde attends a press conference to mark the publication of the 2018 Article IV assessment of the United Kingdom at the Treasury in central London, Monday Sept. 17, 2018.
The IMF said it expected Britain's economy would grow by about 1.5% a year in 2018 and 2019 if a broad Brexit agreement was struck, compared with about 1.75% if it had stayed in.
Prime Minister Theresa May left no doubt that she was prepared to be a "bloody hard woman" in defence of her Brexit plans and intends to reach a "good deal" with the European Union - one that lawmakers would have a choice to accept or see Britain crash out of the bloc.
In today's Observer London Mayor Sadiq Khan warned of the job losses of a badly negotiated Brexit and demanded a new vote for the British people before the United Kingdom cuts ties with the EU.
Meanwhile, a "hard Brexit" Conservative faction says leaving without a deal is preferable to May's proposed terms because it would leave Britain free to strike new trade deals around the world.
A "facilitated customs arrangement", removing the need for customs checks at UK-EU ports.
"Some parts of the draft withdrawal agreement have already been agreed in principle by the United Kingdom and the European Union negotiators, although nothing is agreed until everything is agreed", the note, seen by Reuters, said.
"Fundamental questions - such as the future economic relationship between the two and the closely-related question of the status of the land border with Ireland - remain unanswered", it noted in a statement.
Known as the "People's Vote", the second vote idea is gaining traction among some politicians, unions and business leaders who argue that the public should effectively get a say on the final negotiated deal.
Johnson is the bookmakers' favourite to succeed May, ahead of interior minister Sajid Javid, eurosceptic backbench leader Jacob Rees-Mogg, Gove and Foreign Secretary Jeremy Hunt.
Michel Barnier, the EU's chief Brexit negotiator, is working on a revised plan for Northern Ireland that would involve technology-based solutions to avoid a hard border for goods.
But he told reporters on Monday that a technical fix to the Irish border issue reported by The Times newspaper could allow for a proper Brexit deal to be clinched.
May also insisted no other plan on the table would ensure "frictionless" trade in Ireland. "Barnier is being creative in offering concessions in the text that seem to go the British way but in fact don't change the reality", one European Union diplomat said.