The 21-year-old returned to Liverpool earlier this month having made 20 appearances across all competitions for Tony Docherty's side during the first half of the campaign.
Beck - who was a recent substitute for the Reds in the 4-0 victory over AFC Bournemouth - will now link back up with Dundee until the conclusion of 2023-24 in a bid to gain more senior experience.
The Wales youth international joined Liverpool's Academy set-up at U13 level and has also previously spent time with Famalicao in Portugal and Bolton Wanderers in League One.
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