BoJo's mojo: UK's foreign minister U-turns on Assad
Johnson is expected to make the suggestion at his first meeting as foreign minister in London to discuss the Syrian war with his Western counterparts.
"I will be making clear my view that the suffering of the Syrian people will not end while Assad remains in power. The international community - including Russia - must be united on this," Johnson will say, according to his office.
In March, the former London mayor praised Assad and the Syrian army for their victory over IS in the central city of Palmyra.
Johnson wrote "bravo for Assad" for "saving" the UNESCO World Heritage site, days after Damascus announced that the city had been recaptured from the militants.
"No matter how repulsive the Assad regime may be – and it is - their opponents in Islamic State... are far, far worse," Johnson wrote.
In December, Johnson weighed in on the Syria debate suggesting the West must work with Russian President Vladimir Putin and Bashar al-Assad "to remove an evil death cult" [IS].
Both have been viewed as responsible for killing thousands of civilians in air raids, and putting the war against moderate rebels above the fight with IS.
"Am I backing the Assad regime, and the Russians, in their joint enterprise to recapture that amazing site? You bet I am," Johnson wrote, referencing the ancient Palmyra city.
"That does not mean I trust Putin, and it does not mean that I want to keep Assad in power indefinitely. But we cannot suck and blow at once."
These comments attracted widespread criticism from Syrians, who see Assad and IS as two sides of the same coin.
News that Johnson had U-turned on his previous lukewarm support for the Syrian dictator - responsible for the vast majority of probably hundreds of thousands of civilian dead in the war - prompted a series of responses on Twitter.