The 26-year-old has been identified as a replacement for Calvin Bassey, who is on the verge of joining Dutch giants Ajax.
The Gers have allegedly agreed to sell Bassey to the Eredivisie champions for a club-record fee worth an initial €22m (£18.7m) which could rise to €27m (£22.9m) with add-ons.
Bassey, 22, travelled to Amsterdam on Sunday for his medical ahead of finalising terms on a four-year deal, with Rangers set to make a significant profit on the Nigerian who they bought for around £230,000 from Leicester City in 2020.
According to The Guardian, Rangers are now poised to complete the £4m signing of Davies, who Liverpool bought for around £500,000 in January 2021.
Davies was allegedly on the verge of joining Scottish giants Celtic before receiving a call from Jurgen Klopp, who persuaded the 26-year-old to move to Anfield from Preston North End.
Before his dream move to the Reds, Davies enjoyed a successful seven-year spell with the Lilywhites in League One and in the Championship, making 145 appearances across all competitions.
The Englishman was targeted as a backup option by Liverpool at a time when the Premier League giants had Virgil van Dijk, Joe Gomez and Joel Matip all sidelined with long-term injuries.
However, Davies was never called upon by Klopp and is yet to make a single senior appearance for the club in any competition.
Davies, who spent the 2021-22 campaign on loan with Sheffield United, is now edging closer to moving up to Glasgow where he is set to gain regular first-team football.
Rangers boss Giovanni van Bronckhorst revealed after their 2-1 friendly win over Blackpool on Saturday that the club will need to find a replacement for Bassey ahead of the new campaign.
Speaking to Sky Sports News, the Dutchman said: "The squad is well at the moment. We changed the players who left and got new players in but with Calvin maybe going as well, we have to look for another player to add to the squad.
"We have our list ready. So, once the deal is done we can work on another player coming in."
Rangers have already added centre-back John Souttar to their ranks this summer on a free transfer, while Antonio Colak, Tom Lawrence, Rabbi Matondo and Malik Tillman have also moved to Ibrox, the latter of whom joining the Gers on loan from Bayern Munich.