Moyes, 51, is said to have been drinking at the Emporium wine bar in Clitheroe on Wednesday night when the alleged incident occurred at 10pm.
Lancashire Constabulary said in a statement:
"Police are investigating a report of an assault at the Emporium wine bar in Clitheroe.
"Officers attended the bar at about 10pm yesterday (Wednesday, May 21st) and it was reported that a 23 year old local man had been assaulted by a 51-year-old man. He did not require hospital treatment.
"Inquiries are ongoing and we are speaking to a number of people to try to establish the circumstances.
"There are a number of differing accounts which we need to work through to try to establish exactly what has happened.
"No-one has been arrested at this stage."
Moyes was sacked by United last month, less than one year into a six-year contract.
He presided over United's worst season in 25 years, as the club were ejected from both domestic cup competitions in January and finished seventh in the Premier League table.