Fellow Forest academy products Matty Cash and Joe Worrall were linked with moves to Chelsea (Sun) and Everton (Mirror) respectively over the weekend, and now Osborn is said to be a January target for both Watford and Swansea City.
The report claims the Championship side are braced for offers of around £3 million for Osborn - who has scored one goal and made another five in 20 outings this season - and it adds that the Reds may be tempted to cash in.
Forest tied the 22-year-old to a new four-year deal in August, but a proposed takeover, believed to be led by American businessman John Jay Moores, is yet to go through.
Current owner Fawaz Al Hasawi has sanctioned the sales of several key players throughout his reign, with Scotland winger Oliver Burke the latest to fly the nest having moved to Bundesliga leaders RB Leipzig three months ago.
Osborn has been a Forest regular over the past three campaigns and has been capped by England at Under-18, -19 and -20 levels.
Watford are currently eighth in the Premier League, with 19th-place Swansea in danger of losing their top flight status.