Prime Minister Theresa May has said the BBC should explain why a Jo Brand joke about throwing battery acid was “appropriate content” for broadcast.
The comedian made the remark during a broadcast of Radio 4 satirical show Heresy on Tuesday night.
She was accused by Brexit Party leader Nigel Farage, who has had milkshakes thrown at him by protesters, as “inciting violence”.
In a statement, the BBC said it was “not intended to be taken seriously”.
“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,” the statement – which was first released on Wednesday – continued.
The Metropolitan Police has also confirmed it has “received an allegation of incitement to violence that was reported to the MPS on 13 June”.
It added: “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.”
In the episode, Brand told presenter Victoria Coren Mitchell that people who attacked Mr Farage and far-right political figures with milkshakes were “pathetic”.
She said: “Why bother with a milkshake when you could get some battery acid?,” adding, “I’m not going to do it, it’s purely a fantasy.”
The prime minister’s spokesman said Mrs May has been clear politicians should be able to go about their work and campaign without harassment, intimidation or abuse.
Speaking about the comments on his LBC show on Thursday, Mr Farage said: “This sort of behaviour is completely and utterly disgusting.
‘Plenty of explanations’
“Could you imagine if I was to tell a story like that about somebody on the other side,” he added.
“The police would be knocking on my door within 10 minutes.”
The Sun newspaper said Brand had refused to apologise for the comment after confronting her at her London home earlier.
She is reported to have he added: “I think if they [critics] want an answer there have been plenty of explanations by the BBC and Victoria Coren.”
When asked if she would continue working with the BBC, she is reported to have replied: “I’m not employed by the BBC, so how can they sack me?”