- Video report by ITV News Correspondent Juliet Bremner
Boris Johnson has said "nothing is ruled" out if Londoners disobey social distancing advice from the Government and said he is ready to "do more to ensure" people avoid gatherings.
He told his daily coronavirus press conference there was "some evidence" that in certain parts of London people were not following advice "quite the way we need them to do".
"If we feel that [social distancing advice] isn't working and we need to bring forward tougher measure, nothing is ruled out," he said.
But he said there was "no prospect" of public transport in the capital city being stopped to slow the spread of Covid-19.
He acknowledged there had been some "misunderstanding" over the Government's plans but said people should "avoid gatherings where they transmit the disease".
He added: "We are going to want people to avoid gatherings where they transmit the disease. If it becomes necessary to do more to ensure that, we will certainly do so."
The move fuelled speculation that London was about to be put on lockdown.
But Mr Johnson's official spokesperson rubbished those rumours, telling reporters there was "no prospect" of it.
He dismissed suggestions of a travel ban, telling reporters: "There are no plans to close down the transport network in London and there is zero prospect of any restrictions being placed on travelling in and out of London."
Mr Khan, who earlier took the decision to close dozens of tube stations, said despite the UK being "clearly still in the early phase of this crisis", the capital is "at a more advanced stage".
He told the London Assembly the fast-developing situation means "further measures will need to be introduced at the point at which they will have the biggest effect".
"I can assure Londoners that this is under constant review by the experts," he added.
Rumours of a shut down in London were first sparked by Scotland First Minister Nicola Sturgeon, who has been receiving Cobra emergency briefings.
She said "more stringent" measures could be introduced earlier in London, because it is ahead on the "infection curve".
When asked how far the UK was from a complete lockdown, she said "we know London is ahead of the rest of us".
The Prime Minister later did not rule out following similar moves by other nations in shutting down the city, and stressed he will not hesitate in imposing "further and faster measures".
He said he was reluctant to issue edicts restricting lives, and implored people to follow the advice to socially distance themselves to slow the spread of Covid-19.
"But let's be absolutely in no doubt that these are very, very important choices that we are now making in our daily lives," the PM told his daily coronavirus press conference on Wednesday.
Specifically referencing shutting down London, he added: "We do not rule out - because it would be quite wrong to do so - taking further and faster measures in due course."
London Mayor Mr Khan warned people "should not be travelling, by any means, unless they really, really have to".
"Londoners should be avoiding social interaction unless absolutely necessary, and that means they should be avoiding using the transport network unless absolutely necessary."
He urged Londoners to play their part in preventing the spread of the virus by "avoiding social interaction".
Mr Khan said: "We are in the midst of an immensely challenging and anxious time for all Londoners.
Coronavirus: Everything you need to know:
"We are all rightly worried about our most vulnerable family members, friends and neighbours.
"But London has been through tough times before and throughout our city's history we have repeatedly demonstrated the power of coming together in common cause.
"Now is the time for us to do so again because we all have a part to play in preventing the spread of the virus and helping to save lives."
Mr Khan also suggested transport in the capital could be scaled back even further to help contain Covid-19.
Aside from station closures, the Waterloo and City line and Night Tube services will not run from Friday, but restrictions could go much further.
The Mayor said: "I want to be clear now that the frequency of services is likely to continue to reduce, potentially very significantly, over the days and weeks ahead.
"We will do this in a way that makes sure essential workers can still get around and we will not reduce service levels so that the remaining trains and buses are crowded."