The poll has Ruth Davidson, leader of the Scottish Conservatives, hot on Johnsons's heels with 19 per cent of the vote, followed by high-profile backbencher Jacob Rees-Mogg (17 per cent).
The British newspaper The Telegraph said in a previous report that many had urged British Prime Minister Theresa May to dismiss British Foreign Minister Boris Johnson from office following what they saw as a challenge after publishing a 4,000-word essay in which he envisioned his own vision For Britain's exit negotiations from the European Union (BRIC).
"When members are asked directly to choose who they would like to see as the next party leaders, however, it is Boris Johnson who comes out on top - despite him being the candidate to bring out the largest difference between Remainers and Leavers".
On Friday, he praised a speech by May in which she set out her plan for a roughly two-year transition period after Brexit.
With regards to the single market, Mr Johnson said the United Kingdom should not pay for access after March 2019, and neither should it commit to mirroring European court rulings in exchange for access.
Transport Secretary Chris Grayling also sought to play down any suggestion the Cabinet was not fully behind the Prime Minister.
In an interview with The Sun he said: "I want people to be paid more".
Earlier this month, some of Johnson's cabinet colleagues said he was "backseat driving" on Brexit.More news: UCL: Onyekuru To Earn £35000 Bonus For Anderlecht Win Vs Celtic
But Scottish Conservatives leader Ruth Davidson appeared to take a swipe at the Foreign Secretary.
While taking part in a national day of campaigning, Labour leader Jeremy Corbyn said the latest intervention by the Foreign Secretary demonstrated the Prime Minister was not in control of Brexit strategy.
A new row has broken out in the upper ranks of the Conservatives just as it gears up for its annual party conference in Manchester.
"It seems a pretty dysfunctional government".
"What I have always said is that we will pay for things that are reasonable, scientific programmes".
Meanwhile, Scotland's Europe minister, Alasdair Allan, has said Scotland's relations with Nordic countries will become more important as a result of Brexit.
"So, my determination to act on those concerns, and crucially, to fulfil the promise of my first speech on the steps of Downing Street, is greater than ever".