The 22-year-old midfielder has been a regular in the Championship outfit's first team over the past three campaigns and has been capped by England at Under-18, 19 and 20 levels.
Osborn's form resulted in the club tying him down to a new four-year contract in the summer and he has gone on to enjoy a solid start to this season with one goal and five assists in 20 appearances.
The Hornets have reportedly been keeping tabs on Osborn and could be set to make a £3million bid to try and secure his signature in the January transfer window.
Forest boss Philippe Montanier will be loathe to lose another young starlet from his ranks after Oliver Burke was allowed to leave for German Bundesliga leaders RB Leipzig on transfer deadline day in the summer.
Watford's hopes of signing Osborn could rest on a proposed takeover at the City Ground with American businessman John Jay Moores looking to buy the club.
If that goes through before January it could significantly damage the chances of a sale as Moores will no doubt look to make an early impression on the Forest faithful by keeping their top player, while current owner Fawaz Al Hasawi is not afraid to sell as he has proven in the past.