The England international is expected to leave the Foxes ahead of the 2020-21 campaign, and both Chelsea and City have been credited with an interest in his services.
However, according to The Sun, United are ready to 'battle it out' with their Premier League rivals for the left-back, with Red Devils boss Ole Gunnar Solskjaer determined to build a title-winning side.
The 23-year-old has been in impressive form for Brendan Rodgers's side during the 2019-20 season, scoring three times and providing four assists in 33 appearances in all competitions.
Leicester are believed to value Chilwell in the region of £60m and are in a strong position when it comes to negotiations as they have the defender on a contract until June 2024.
Luke Shaw is currently United's first-choice left-back, and the former Southampton youngster is arguably enjoying his most productive season for the club since arriving in 2014.