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Tsimikas has spent five years at Liverpool acting as a back-up to first-choice left-back Andy Robertson.
However, the Reds spent £40million to bring in Milos Kerkez from Bournemouth this summer and he started in the Community Shield and Liverpool's Premier League opener.
The report adds that Nottingham Forest are also interested in Tsimikas, while Leeds previously considered a move for the full-back.
When asked about his future ahead of the final game of last season, Tsimikas told Sport 24 : "It's definitely a temptation to play more.
But for me, the 27-29 games I play every year in all competitions are more than playing 40 for another team and in another league.
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"Here I am on the Broadway of football.
I'm happy here, for five years I've always been happy with these players I had by my side and with the jersey I wore.
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"It's very difficult to reach such a level, to play in 27 games in a season with a Liverpool shirt.
I would definitely like to play more, but from then on in this league, with this team I was always happy and I always gave my best, so I was OK with the games I played."
Marseille are interested in Tsimikas(Image: Liverpool FC via Getty Images)
Liverpool have enjoyed a busy summer so far, spending more than £300m on Florian Wirtz, Hugo Ekitike, Milos Kerkez, Jeremie Frimpong, Giovanni Leoni, Giorgi Mamardashvili, Armin Pecsi and Freddie Woodman.
They have also raised almost £200m through player sales, with Luis Diaz, Darwin Nunez, Jarell Quansah, Ben Doak, Caoimhin Kelleher, Tyler Morton, Trent Alexander-Arnold and Nat Phillips all departing.
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