The idea for the 22-mile-long crossing was reportedly put forward during an Anglo-French summit that took place on 18 January 2018, according to publications including The Guardian.
During the meeting, which was called to discuss issues including Brexit and immigration, Johnson is understood to have described the fact that the two countries are connected only by the undersea Eurostar rail tunnel as "ridiculous". French president Emmanuel Macron agreed, saying "let's do it", according to The Telegraph.
Johnson, who played a lead role in the campaign for the UK to leave the European Union in the 2016 EU referendum, tweeted: "I'm especially pleased we are establishing a panel of experts to look at major projects together."
"Our economic success depends on good infrastructure and good connections. Should the Channel Tunnel be just a first step?" he continued. If the project were to get the go-ahead, the 35.4 kilometre structure would be in the top 20 of the longest bridges in the world.
Industry figures ridicule plans for "Boris bridge"
But the foreign secretary's suggestion has largely been met with scepticism and ridicule. The UK Chamber of Shipping tweeted: "Building a huge concrete structure in the middle of the world's busiest shipping lane might come with some challenges".
The Guardian's architecture critic Olly Wainwright jibed: "What are the odds that Thomas Heatherwick has already designed it?", referring to Heatherwick's abortive Garden Bridge, which Johnson commissioned while mayor of London. Meanwhile architect Alan Dunlop told The Times it would be easier and less expensive to "just move France closer".
Business Insider's political editor Adam Bienkov remarked: "A reminder that Boris Johnson's previous plans for public money have included an island airport in a birdstrike zone, the most expensive footbridge in history and a bus that was so hot inside it doubled up as a mobile sauna."
But Ian Firth, bridge engineer at COWI and former President of the Institution of Structural Engineers, has said that he believes building such a bridge is "entirely feasible".
"It has been looked at before," he said in an interview on BBC Radio 4's Today programme. "Before the tunnel was built there were bridge options being looked at and of course it is doable."
But Downing Street has since down-played rumours surrounding the bridge, saying that there are "no specific plans" and that Johnson had simply proposed the idea of a "fixed link".
Johnson isn't the first to propose an above-sea link between the two countries – transport officials submitted plans for one in 1981.