Silva, 26, joined the Istanbul club on a season-long loan last September after a late move to Sporting Lisbon collapsed on transfer deadline day.
The switch to Turkey brought limited opportunity — he has managed just 619 minutes across all competitions, scoring four goals, with a December injury disrupting any momentum he had built in the Süper Lig.
Turkish outlet Fanatik reported last month that Silva is "definitely leaving" Beşiktaş, and the Daily Mail reports that he has made clear to his representatives that he has no desire to return to Turkey. His preference is for a move to another European league, with options in England and on the continent being assessed.
The winger joined Nottingham Forest from Vitória Guimarães in the summer of 2024 for €7 million. He was popular with teammates and supporters during his one season at the City Ground, registering five goals and three assists across 38 appearances, but was deemed surplus to requirements when the club reshaped their squad ahead of this campaign.
Forest themselves are in the middle of a remarkable run. They beat Chelsea 3-1 at Stamford Bridge on Monday to move six points clear of the Premier League relegation zone, and face Aston Villa in the Europa League semi-final second leg at Villa Park on Thursday with a 1-0 advantage from the first leg at the City Ground.
Manager Vitor Pereira described Morgan Gibbs-White's chances of playing as a miracle after the playmaker sustained a severe facial injury in the collision with goalkeeper Robert Sanchez during Monday's match.
Whether Forest retain, sell or loan Silva will be among the quieter decisions of what promises to be an eventful summer at the club.