The 27-year-old has been with Santos in his native Brazil for the last four years but his current deal is due to expire in December, meaning that he will be a free agent by the time the window re-opens.
Lima's agent Luccas Badia has claimed that United are among a series of top European clubs - along with Barcelona and Inter Milan - to have made enquiries about the player, who has 10 goals in 107 league appearances for Santos.
Badia has also alleged that Crystal Palace failed with a €15m (£13.2m) bid over the summer, while an even better offer of €20m (£17.6m) from Fenerbahce was also turned down.
"Inter, Roma and Barcelona are keeping close tabs on Lucas, Manchester United want him," the agent is quoted as saying by Turkish newspaper Haberturk.
"I think we would be well suited to the Serie A but we will see. His contract ends on 31 December and there are many clubs interested but I am not sure what Santos is trying to do.
"Crystal Palace made a €15 million offer while Fenerbahce made a €20m bid but for some reason neither has been accepted."