Video report by ITV News Correspondent Martha Fairlie
A comment made by Jo Brand on BBC Radio 4 programme Heresy about throwing battery acid at politicians is being assessed by the Metropolitan Police following an allegation of incitement to violence, the force said.
The BBC earlier defended the comedian against claims she encouraged violence in a joke about throwing battery acid during a radio show.
Brand, who joined the Heresy's show on Radio 4 as a guest on Tuesday, joked about throwing battery acid at "unpleasant characters" instead of milkshakes.
A police spokesperson 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 inquiries are ongoing."
Nigel Farage, leader of the Brexit party, who had a milkshake thrown on him recently hit back at the comedian on Twitter saying: "This is incitement of violence and the police need to act."
When asked if Brand would continue working with the BBC on Thursday, she responded: "I'm not employed by the BBC, so how can they sack me?"
A BBC spokeswoman said the jokes made on the Heresy show are "deliberately provocative and go against societal norms".
She added they are "not intended to be taken seriously."
Downing Street has said the BBC should explain why it broadcast the comedian's comments.
The Prime Minister's official spokesman 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."
Social media users have compared the BBC's response with Danny Baker's sacking over an alleged racist tweet about royal baby Archie.
On the show, Brand said: "Why bother with a milkshake when you could get some battery acid?"
The 61-year-old then made clear she was joking and criticised the milkshake stunts.
She said: "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," she added.
After the show, presenter Victoria Coren Mitchell said she hoped Brand's remarks had not caused offence but added the aim of her show was to "test the boundaries of what it's okay to say and not say".
The TV personality later responded to Mr Farage on Twitter, accusing him of double standards.
She wrote: "Nigel! I'm genuinely disappointed; we don't agree on everything, but I would totally have had you down as a free speech man. Especially when it comes to jokes."