The centre-back's long-term future on Merseyside has long been uncertain and a handful clubs from countries such as Germany, Spain and even Greece are monitoring him.
The Portugal Under-23 international played for his country's men's football team at the Rio 2016 Olympics in Brazil earlier this month and his fine performances have attracted interest from Olympiakos, as well as Eintracht Frankfurt, Freiburg and Granada.
23-year-old Ilori joined from Sporting Lisbon in the summer of 2013 but failed to make a senior appearance for the club after his £7 million move.
Loan spells at Granada, Bordeaux and Aston Villa - none of which were particularly successful - all followed before his first competitive appearance for Liverpool.
But he made just three appearances last season, the only three of his Liverpool career so far, and all of those came in cup competitions.
Ilori nearing closer to Liverpool exit
Although the Reds' defence still has glaring weaknesses, as showcasd in their first two Premier League games of the season so far, Klopp doesn't envisage any role for Ilori.
He is behind Dejan Lovren and Ragnar Klavan, who partnered each other against both Arsenal and Burnley, as well as Mamadou Sakho, summer signing Joel Matip and Joe Gomez - yet to return from a long-term injury lay-off - in the centre-halves pecking order.
And although he is back with Liverpool after the conclusion of the Olympics, his stay at Anfield will finally come to an end before the end of the transfer window on August 31.
Ilori, one of several back-up players Liverpool remain determined to offload, could seal a move to reigning Greek champions Olympiakos. However, it is thought they prefer a loan move rather than a permanent deal.
Liverpool are looking for around £3 million to let Ilori go and know of interest from aforementioned Bundesliga sides Eintracht Frankfurt and Freiburg, as well as a potential return to La Liga side Granada.
Ilori is one of several Liverpool players who could yet leave the club before the closure of the window, with Luis Alberto and Mario Balotelli two of those without futures at the club.
Lazar Markovic and Andre Wisdom are among the others who have question marks hanging over their heads, having featured in pre-season but dropped off the radar more recently.
The £32 million sale of Christian Benteke to Crystal Palace last week means Liverpool have recouped around £65 million in fees, with add-ons potentially increasing that fee, and boast a positive net spend.