A spokesman for Cheshire Constabulary confirmed the incident had been reported.
One witness told the MP: “Leave it,” before another was heard to say: “The MP of gobs—–.”
Another onlooker then said: “He’s the local MP,” to which Mr Amesbury replied: “Yes, I am. And you’ll never threaten me ever again.”
He added: “Come and threaten the MP, you’ll never do it again, will you? You’ll never threaten the MP again, will you? You soft f—ing lad.”
Mr Amesbury was first elected in 2017 and previously served in Jeremy Corbyn and Sir Keir Starmer’s shadow cabinets.
He was not given a ministerial role when Labour won the general election in July.
The spokesman for Cheshire Constabulary said: “At 2.48am on Sat Oct 26, police were called to reports of an assault in Frodsham.
“A caller reported he had been assaulted by a man in Main Street. Enquiries are ongoing.”
A Labour party spokesman said: “We are aware of an incident that took place last night. We understand that Mike Amesbury MP approached Cheshire Police to report what happened this morning himself and that he will co-operate with any inquiries they have.”
Labour has not suspended Mr Amesbury for the incident.
Mr Amesbury was approached for comment.