Forest have been in contact with Saelemaekers' representatives for over a year, even before his loan move to Roma last season.
The 25-year-old attacker spent the 2024/25 campaign on loan at Roma, where he overcame an injury-disrupted start to play 31 games, scoring 7 goals and providing 7 assists.
His versatile attacking abilities have caught the eye of Forest's manager Nuno Espirito Santo, who views Saelemaekers as a potential replacement and cover for Anthony Elanga, the winger currently linked with a move to Newcastle United.
AC Milan are asking for a fee in the region of 25 million euros, while Forest value the player slightly lower at around 20 million euros. This difference remains a key hurdle, but sources suggest it could be bridged depending on negotiations and Allegri's assessment.
Milan's new manager, Massimiliano Allegri, has paused all transfer discussions to evaluate Saelemaekers during pre-season before making a final decision on his future.
Meanwhile, Eintracht Frankfurt has recently emerged as a serious contender for the Belgian's signature.
Newcastle United previously sought information on Saelemaekers earlier this year but did not progress talks, as their interest shifted towards Elanga.
Newcastle are preparing a renewed attempt to sign Nottingham Forest winger Anthony Elanga, with reports suggesting a swap deal involving Harvey Barnes is being considered.
Fiorentina have decided against activating their 17 million euro option to sign Albert Gudmundsson permanently from Genoa, but the Viola remain keen to keep the Icelandic star.
Nottingham Forest have secured the long-term services of head coach Nuno Espirito Santo, handing him a contract extension that runs until 2028.
The Premier League fixtures for the 2025/26 season have been confirmed, promising excitement and intrigue as clubs fight for ascendancy in England's top flight.
