The Reds have pursued the 21-year-old Sevilla left-back all summer and had a £16 million bid rejected back in May.
The La Liga side were said to be holding out for around £20 million for their defender, and the Reds have now conceded that a deal for the player is out of the question.
Guilherme Siqueira - who was a target in the summer of 2013 - has moved to Atletico Madrid, while Chelsea are said to be reluctant to allow Ryan Bertrand to move to a Premier League title rival.
Davies, who Rodgers worked with during his time in Wales, has now emerged as the top target as the Northern Irishman looks to bring an end to a position that has plagued the Reds for years.
First choice left-back Jose Enrique spent much of the season on the sidelines, and Aly Cissokho's inability to adapt to Premier League football forced Rodgers to play Jon Flanagan out of position for the final few months of the campaign.
Flanagan's adaptation was admirable, and the youngster won plenty of plaudits for his no-nonsense approach, but Rodgers has been keen to land a top-class specialist left-back throughout this summer.
Moreno was top priority, but Sevilla's firm stance has left the Anfield outfit with no other choice but to pursue alternatives.
It has been a busy transfer window so far for the Reds, with around £60 million already splashed out on the likes of Adam Lallana, Rickie Lambert, Emre Can and Lazar Markovic - who joined for £20 million from Benfica on Tuesday afternoon.
Wilfried Bony is said to be on Rodgers' radar, while speculation linking the Reds with Southampton's Dejan Lovren continues to swirl.