Downing Street has said there are "no specific plans" for a bridge between the UK and France after Boris Johnson floated the idea of a "fixed link".
The UK foreign secretary reportedly ran the idea past French President Emmanuel Macron at a summit on Thursday.
But Downing Street said he had been referring to a panel looking at Anglo-French major projects.
Experts said a bridge was technically possible but some critics ridiculed the concept.
Thursday's summit with Prime Minister Theresa May at Sandhurst's Royal Military Academy in Berkshire was Emmanuel Macron's first UK visit since becoming French president.
The BBC understands Mr Johnson made the comments while discussing other major infrastructure projects around the world, including a bridge in Japan.
Mr Johnson said in a tweet that the UK and France had agreed that a panel of experts should be assigned to look at "major projects".
A source close to the foreign secretary said he also thought it was "crazy" the two countries were only connected by one railway line when they were merely 20 miles apart.
Asked about the remarks, the prime minister's spokesman said: "I haven't seen any plans on that. We are going to have very close ties with France economically, culturally and in areas such as defence and security for many many decades to come.
"What was agreed yesterday (Thursday) and what the foreign secretary tweeted about as well is a panel of experts who will look at major projects together, including infrastructure. And we want to work very closely with our French colleagues on building a shared, prosperous future."
'Not a fan'
Home Secretary Amber Rudd, on a visit to Crawley, West Sussex, was more forthright: "I think I'm safe saying that I am not a huge fan of this latest proposal."
A spokesman for the French president confirmed he had had a conversation with Mr Johnson about a bridge and said Mr Macron had told him: "The issue of access is an important one." But he denied reports the French president had responded with "I agree, let's do it".
It is not the first time the idea of a Channel bridge has been raised - transport officials submitted plans for one in 1981.
The proposals were dismissed as "impractical" at the time, as it would make it difficult for ships to navigate the waters.
Speaking to BBC Radio 4's Today programme, bridge designer and former president of the Institution of Structural Engineers Ian Firth said it would be a "huge undertaking, but absolutely possible".
But the idea was greeted with scepticism on social media. Labour's Emily Thornberry tweeted that she had assumed it was "not real".
Boris's big projects
Boris Buses: As mayor of London, Mr Johnson launched a competition in 2008 to design a new London bus to replace the iconic Routemaster. They hit the roads in 2012 but faced criticism over their cost, problems with their hybrid engines and non-opening windows. Transport for London have said they are not buying any more.
Emirates Air Line: The £60m cable car over the Thames near the O2 Arena opened ahead of the London Olympics in 2012, again during Mr Johnson's tenure as London mayor. He said it was an ideal way of linking north and south London but it faced criticism over its cost and limited passenger numbers.
Boris Island: Long backed by Mr Johnson as an alternative way of expanding airport provision in south-east England, the £47bn plan for an island airport in the Thames Estuary was rejected as an option by the commission looking into the issue in 2014. He had argued it could "reshape the economic geography" of south-east England.
Garden Bridge: A £200m plan, backed by Mr Johnson, to build a bridge covered with trees over the River Thames in central London was abandoned last year, having lost the support of his successor as mayor of London, Labour's Sadiq Khan.
Dover MP Charlie Elphicke backed Mr Johnson's idea, saying: "Boris is right. We absolutely must invest in infrastructure to keep trade flowing between Britain and France." But former UKIP Nigel Farage told the BBC that the idea sounded like "a big waste of money".
Former Lib Dem leader Sir Nick Clegg told BBC Radio 4's Westminster Hour the idea was "a masterstroke in diversion".
"It's all very well talking about a bridge that may or may not be built in decades to come. The thing that worries me much more is he wants to pull up the drawbridge to the economic relationship which has served us so well over such a long period of time."
Conservative MP Mark Pritchard was not a fan of the idea, tweeting it was "a bridge too far":
An overground “fixed link” May be good for trade between UK & France but such an idea forgets how the English Channel has saved the UK from invasion many many times. It would also be an illegal migrants Yellow Brick Road.....A bridge too far.— Mark Pritchard MP (@MPritchardUK) January 19, 2018
Some on social media used the contentious £350m claim used by Vote Leave during the EU Referendum campaign against Mr Johnson or compared it to Donald Trump's famous campaign promise to build a wall between the US and Mexico.
Boris Johnson: ‘we’re going to build a beautiful bridge, it’s gonna be beautiful folks. And France is gonna pay for it! #BorisBridge— Steven Maguire (@smaguire4621) January 19, 2018
"We will build a bridge to France! Building bridges, yes. A bridge to Ireland, bridges to, to, to Madison County, a bridge over the river Kwai, a Fifth Bridge, a Sixth Bridge, a Bridge to Terabithia, a Jeff Bridges, a, a, a, a Brigit Jones Diary. Glorious. Yes." #BorisBridge— We Need EU 🇪🇺 (@WeNeedEU) January 19, 2018