Middlesbrough say the Ghanaian winger is not for sale at any price.
Forest are said to have lodged a £1.75million bid for the 26-year-old, but Boro made it clear that the player is not for sale under any circumstances.
Adomah was Middlesbrough's top scorer last season. He netted 12 times as the club finished the Championship season in 12th place, safely in mid-table.
This summer he received a call-up to the Ghana squad for the 2014 World Cup, but saw only 12 minutes of action as the Black Stars were knocked out in the group stage.
He is reportedly one of Stuart Pearce's top targets this season as the new Forest boss looks to build a side capable of gaining promotion to the Premier League.
Gazette Live reports, however, that Middlesbrough are determined to hold on to their star man, having already lost striker Marvin Emnes, and Forest's bid was rejected "emphatically".
Adomah has also been linked to Crystal Palace this summer, and a number of Premier League teams are said to be interested in securing his signature.