Ms Davidson, who along with Mr Johnson is one of the favourites to succeed Mrs May as Tory leader, said that "overoptimism" about the future of Britain outside the European Union "sells people short" - a direct reference to Mr Johnson's 4,000-word Brexit essay in The Daily Telegraph a fortnight ago.
He also said the final post-transition deal must not see the United Kingdom just copying Brussels rules to ensure easy access to the single market.
Johnson's resurgent performance in Conservative leadership polling suggests his gamble on making a Brexit intervention has paid off.
In an interview with The Sun he said: "I want people to be paid more".
Meanwhile, Scotland's Europe minister, Alasdair Allan, has said Scotland's relations with Nordic countries will become more important as a result of Brexit.
Do I want the delay to go on longer than two years?
Some of Johnson's cabinet colleagues recently accused him of "backseat driving" on Brexit after setting out his vision for the UK's future outside the European Union in a 4,300-word newspaper article just days before a key speech by the prime minister.
During that speech, Mrs May announced a two-year transition period when the United Kingdom leaves in 2019.
"If you're not prepared to walkaway you're just a pushover and they dictate to you".
Mr Johnson said the minimum wage - at £7.50 an hour rising to £9 in 2020 - was not enough.
Johnson's call comes as other Eurosceptic Tory MPs have said the United Kingdom should walk away from Brexit talks by Christmas unless the grouping shows it's serious about a free trade deal.
Signatories calling for the hardline stance include former Brexit minister David Jones, and the issue is set to feature heavily at the Tory conference.
The data also shows that most Conservatives want May to resign before the next general election, despite the prime minister stating she wishes to fight the next election.
Ms May has said the transition period must last around "about two years".
The Conservative Party conference takes place in Manchester from Sunday until Wednesday.
"But there is the idea that this is quite hard and it needs serious people to do a lot of legwork and scan the detail to make sure we do get to a place where it will all be OK". Conservatives have a plan to make that a reality.