A comment made by Jo Brand about throwing battery acid at politicians is being assessed by the Metropolitan Police following an allegation of incitement to violence.
The comedian made the joke on Victoria Coren Mitchell's Heresy radio programme on Tuesday night, and was criticised by Brexit Party leader Nigel Farage among others.
A police statement to the Press Association on Thursday said: "Police have received an allegation of incitement to violence that was reported to the MPS on 13 June.
"The allegation relates to comments made on a radio programme. The allegation is currently being assessed.
"There have been no arrests and enquiries are ongoing."
The Press Association understands that the allegation reported to the police was not made by Mr Farage or the Brexit party.
In a tweet on Wednesday, Mr Farage accused Brand of inciting violence, although he did not say who against.
He wrote: "This is incitement of violence and the police need to act."
Scotland Yard said no arrests had been made, but that inquiries were ongoing.
On Thursday, Theresa May's official spokesman said the BBC should explain why it broadcast her comments.
He said: "The Prime Minister has been repeatedly clear that politicians should be able to go about their work and campaign without harassment, intimidation or abuse.
"I note that Brendan Cox has said that violence and intimidation should not be normalised and we should consistently stand against it.
"The Prime Minister shares this view. It is for the BBC to explain why it considers this to have been appropriate content for broadcast."
Last month, Mr Farage was showered with milkshake during a campaign walkabout in Newcastle city centre.
Broadcasting watchdog Ofcom said it has received 65 complaints about the episode of Heresy.
On Wednesday, the BBC defended airing Brand's remark, but the corporation has not commented again.
A BBC spokeswoman said: "Heresy is a long-running comedy programme where, as the title implies and as our listeners know, panellists often say things which are deliberately provocative and go against societal norms but are not intended to be taken seriously."
In reply to a question about the state of UK politics, Brand had told the programme: "Well, yes, I would say that but that's because certain unpleasant characters are being thrown to the fore and they're very, very easy to hate and I'm kind of thinking 'Why bother with a milkshake when you could get some battery acid?'
"That's just me. I'm not going to do it, it's purely a fantasy, but I think milkshakes are pathetic, I honestly do, sorry."
At the end of show, Coren Mitchell said she hoped Brand's remarks had not caused offence but added that the radio series had been set up to "test the boundaries of what it's OK to say and not say".