The right-back has made just five appearances all season after an injury ravaged season last year.
He was not involved in Thursday's defeat at the Etihad, with the Reds agreeing a loan deal with Bournemouth.
Nathaniel Clyne heading to Bournemouth on loan. #LFC
The 27-year-old's move has not yet been confirmed by the club and it would be a surprise to see the manager let go off another valuable option.
However, it could mean that the boss is comfortable with the situation due to having Fabinho in the squad.
The Brazilian is a regular right-back for his national team and started his time at Monaco playing in the back-line.
Clyne has made 103 appearances since moving from Southampton in 2015 and had been a reliable option under Jurgen Klopp before his injury.
Eddie Howe's club beat off competition from Brighton to agree a permanent deal with Liverpool, with the youngster expected to have joined Crystal Palace on loan earlier this week.