Ayew is keen to reunite with Forest manager Steve Cooper, who he worked under at Swansea in the Championship.
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The 33-year-old is a free agent after ending his stint at Qatari side Al Sadd, making him available after the close of the January transfer window.
Everton were said to be keen on bringing him in to strengthen their attacking options after a winter window that saw them become the only Premier League side not to make a signing.
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The Toffees thought they had secured Netherlands winger Arnaut Danjuma on loan from Villarreal, but that deal collapsed when he headed to Tottenham, despite passing a medical with the Merseyside squad.
Everton are currently in a relegation battle, sitting 19th in the table with their three victories the fewest of any side.
Frank Lampard was relieved of his duties earlier in January after a poor start to the season, seeing long-time Burnley boss Sean Dyche named as his successor.
Dyche has also been linked with five-time Champions League-winner Isco as another free agent signing as he looked to reverse the club's downward trajectory.
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That won't have been helped by getting beaten to a player by Forest, though, with the newly-promoted side six points and six places above them in the Premier League.