Adomah is currently still away on his summer break after being at the World Cup finals with Ghana in Brazil, where he featured in one match against USA.
The 26-year-old only joined Boro last summer in a £1million deal from Bristol City and he enjoyed a hugely successful campaign with 12 goals in 40 league games.
Sky Sports understands a number of Premier League clubs are preparing offers, but it is Forest who have made the first move.
Forest are understood to have made their opening offer last week, but Boro were quick to turn down the bid.
Boro, who have just spent nearly £3million on Spanish striker Kike, have already sold Marvin Emnes and are set to offload Lukas Jutkiewicz to Burnley, so do not need to part with Adomah.
However, should Premier League interest materialise as expected, it may prove harder to keep Adomah at The Riverside.