The Brazilian signed for Saudi Pro League club Al Ahli in 2023 after eight seasons on Merseyside and helped his side to a third placed finish.
This season, Firmino has netted six goals in 19 games but the future of the Al Ahli captain could lie away from Saudi Arabia.
They are now open to offers for the 33-year-old as they look to bolster other areas of their squad.
The club are looking to recruit both a left winger and a holding midfielder in their hopes of improving on their fifth-place position in the table.
Firmino's inclusion in the squad currently takes up one of the ten spots allowed in the foreign player quota, although eight are permitted to be used in any matchday squad.
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Mourinho has been keeping a close eye on Firmino as part of his own title charge in the Turkish Super Lig and could make a move for the Brazilian in the January transfer window.
Fenerbahce sit six points adrift of city rivals Galatasaray and hold high hopes of closing the gap after the winter break.
However, the Istanbul club are unlikely to get a free run at the Brazilian with fellow Saudi club Al Tauwoun interested in taking him on loan.
There has yet to be any former approaches for Firmino but Al Ahli are open to any offers for the former Liverpool man.
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Should Firmino depart this winter, he could be the second former Reds star to swap Saudi Arabia for Europe after Jordan Henderson's switch from Al-Ettifaq to Ajax.
Firmino made 362 appearances for the Reds, scoring 111 times after signing from Bundesliga outfit Hoffenheim in 2015.
He formed part of one of the Premier League's most fearsome three-pronged attacks alongside Mohamed Salah and Sadio Mane.
Firmino played a key role in Liverpool's Champions League triumph over Tottenham in 2019, Premier League title success in 2020 and FA Cup, Carabao Cup and FIFA Club World Cup wins.
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