Johnson made the radical suggestion during Thursday's Franco-British summit near London, a source in the French presidency said, prompting reactions ranging from polite diplomatic putdowns to bemusement to scorn.
The Daily Telegraph reported him telling the French leader that it was "ridiculous" that the neighbours were "linked by a single railway".
Prime Minister Theresa May's office played down the idea on Friday, with a spokesman saying: "I haven't seen any plans on that."
French Finance Minister Bruno Le Maire, speaking to France's Europe 1 radio, seemed unconvinced but did not reject the proposal outright, saying: "All ideas merit consideration, even far-fetched ones."
A bridge would have to span the 22-mile (35-kilometre) channel between Dover, on England's south coast, and Calais in northern France, but shipping experts wondered what knock-on effects it would have.
"Building a huge concrete structure in the middle of the world's busiest shipping lane might come with some challenges," the UK Chamber of Shipping said in a tweet.
Britain and France are currently linked by the 32 mile- long (50.5 km) Channel Tunnel, which carries high-speed Eurostar passenger trains, the Eurotunnel Shuttle for road vehicles and international goods trains.
In 2016 it carried over 10 million passengers, but only operates at 54 percent capacity, according to a Eurotunnel spokesman.
It is not the first time that Johnson has raised the idea of a major infrastructure project.
While London mayor, he called for an airport to be built in the Thames Estuary to relieve congestion at Heathrow, and his plans for a garden bridge in central London were dashed by his successor Sadiq Khan.
Macron scored a diplomatic coup during his trip by agreeing to loan the Bayeux Tapestry, an embroidery that dates from around 1077, to Britain in 2022.
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