The 21-year-old returned to Liverpool FC 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 FC Dundee until the conclusion of 2023-24 in a bid to gain more senior experience.
The Wales youth international joined Liverpool FC’s Academy set-up at U-13 level and has also previously spent time with Famalicao in Portugal and Bolton Wanderers in League One.