The Dutch international, 20, was linked with both of the Premier League rivals last summer, before ultimately leaving Ajax for Bayern in a £21m move.
But Gravenberch's first season with the Bundesliga giants has been far from plain sailing – he's made just four starts in all competitions – and Jurgen Klopp is said to be keen to add him to the Reds' midfield ranks.
To that end, Liverpool appear to have already set the wheels in motion: according to Christian Falk of Bild, the Reds have met with Gravenberch's agent about a possible switch to Anfield.
Falk added that Bayern are insisting Gravenberch – who Transfermarkt currently value at €30m (£26.8m) – is not for sale, but German outlet Fussball Transfers claim that Liverpool haven't been deterred by that stance and are determined to get a deal done.
Gravenberch has made 18 appearances for Bayern – but 17 of those have come from the bench, and he's featured for only 27 minutes combined under new boss Thomas Tuchel.
The early signs are that he won't play a major part in Tuchel's plans going forward, so he may well try to push for an exit in the next transfer window.
Arsenal have also been linked with Gravenberch, who has won 22 caps for the Netherlands – although his lack of playing time for Bayern cost him a spot in his country's 2022 World Cup squad.
Barring an unfathomably disastrous end to the campaign, the Gunners would be able to offer Gravenberch Champions League football in 2022/23 – while Man United are well on track to return to the competition too.
Liverpool, on the other hand, look set to have to settle for a Europa League place at best – although, given his nightmare time at Bayern so far and the crucial developmental stage of his career he's at, his main consideration simply has to be playing more regularly.
More Liverpool stories
Liverpool are being linked with players all over Europe ahead of what could be a busy summer of business at Anfield: Josko Gvardiol, Aurelien Tchouameni and Nicolo Barella are among the club's potential targets.
Meanwhile, former Red Sadio Mane has been tipped for a quick Premier League return with Manchester United– after coming to blows with Bayern Munich teammate Leroy Sane after their Champions League defeat to City earlier this week.
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