Arsenal manager Arsene Wenger confirmed his interest in bringing the former Borussia Dortmund man to north London last week and it had been reported that Sahin preferred a move to a Champions League club.
However, following his side's 1-0 victory over Hearts on Thursday, Rodgers seemingly admitted that the Reds have trumped their Premier League rivals to his signature.
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"It's quite far [along the line]. He's a very good footballer."
The Turkish midfielder struggled to break into Jose Mourinho's first-team plans last season, making just four league appearances in his first campaign in La Liga.
The Spanish side moved to secure his services following his contribution to Borussia Dortmund's victorious 2010-11 Bundesliga campaign.