The 20-year-old's form in Germany has drawn interest from a number of Europe's top clubs, with initial rumours linking him with a January move to England.
A winter switch never materialised, but the winger - who has scored eight goals in 18 Bundesliga appearances - has previously expressed his desire to play in the Premier League.
United, Chelsea and Arsenal are among the clubs thought to be monitoring his progress, but German publication Kicker reports that Leverkusen will demand nearly £100m for his signature.
Bailey's current contract with the German outfit runs until 2022, and they are understood to consider him of similar value to Ousmane Dembele, who joined Barcelona from Borussia Dortmund for £97m.
Bailey only joined Leverkusen from Genk in January of last year, with the Bundesliga club paying around €20m (£17.75m) for his services.