It is understood Robertson was not pushing for a move, although the Scotland captain is keen to play more regularly before this summer’s World Cup.
Liverpool signed him from Hull City for £8m in 2017 during Jurgen Klopp’s reign and he has made 364 appearances for the Reds.
The 31-year-old has won nine trophies during his time at Anfield, including two Premier League titles and the Champions League.
However, he is no longer the first-choice left-back under Arne Slot, with Milos Kerkez the preferred option.
Robertson came off the bench to replace Kerkez against Bournemouth but he has started just four times in the Premier League this season.
After the match, Liverpool captain Virgil van Dijk said he wants Robertson to stay, calling him a “very important” member of the squad.
Defending champions Liverpool are sixth in the league, 14 points behind leaders Arsenal, but are in a strong position to reach the Champions League last 16 and will face Brighton in the fourth round of the FA Cup next month.
The transfer window closes on 2 February.