The comedian made the joke on Victoria Coren Mitchell’s Heresy radio programme on Tuesday, and was criticised by the Brexit Party leader, Nigel Farage, among others.
Scotland Yard confirmed it received an “allegation of incitement to violence” on Wednesday.
A police statement said: “Police have received an allegation of incitement to violence that was reported to the Metropolitan Police Service 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.”
During the radio show, the comedian was asked about the “terrible” state of British politics.
She replied: “Well, yes I would say that but I think 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?’”
As the studio audience laughed, the 61-year-old added: “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.”
Mr Farage, who was hit by a milkshake in Newcastle city centre during the European parliament election campaign last month, subsequently denounced the comments on Twitter.
He wrote: “This is incitement of violence and the police need to act.”
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.”
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.”