According to Chris McKenna of the Daily Star, the Reds will go head-to-head with their Premier League rivals for the youngster after protracted talks to sign Alberto Moreno from Sevilla finally broke down.
McKenna claims Liverpool had an £8 million bid for Davies rejected on Tuesday, but will return with an improved offer to trump the £10 million deal Tottenham agreed in principle on Wednesday.
A deal with either club is expected to go through next week.
Former Swansea boss Rodgers worked with Davies during his two seasons at the Welsh club from 2010 to 2012, although the 21-year-old was still in the youth ranks at that time.
The defender made his debut in August 2012 under Rodgers' successor Michael Laudrup and became a regular starter that season, following an injury to first-choice left-back Neil Taylor.
The Academy graduate went on to win the League Cup and a first cap for Wales in that same season.
The full-back is clearly a rising star. During the 2013-14 season he made 41 appearances in all competitions for Swansea, per Soccerbase. That stint included Europa League football, where he faced the likes of Napoli.
Given his experience, relative to his age, Davies would certainly be an excellent option for Rodgers, who has been hunting for a left-back all summer.
Last season the Reds were left short in that position following an injury to Jose Enrique. Aly Cissokho failed to impress during his season-long loan from Valencia, so Rodgers turned to right-footed full-back Jon Flanagan, 21, to fill in on the left of the defence.
Davies's arrival at Anfield would free up Flanagan to challenge Glen Johnson in his preferred right-back role, which is something many Liverpool fans would love to see happen.
Welshman Davies is a product of the Rodgers school of football and would easily slot into a Liverpool side that play with the same passing and possession philosophy as the Swansea team he left behind.
Davies's stats would bear that out. His pass completion of 84 percent is slightly higher than both Flanagan (83 percent) and Enrique (80 percent) last season. Defensively his successful tackles per game (1.74) dips below Flanagan’s 2.57, but in terms of interceptions (1.85), blocks (0.32) and clearances (3.82) he outperforms his two rivals, per Squawka.
And, irrespective of the fee, it could be argued that Davies is a better option than Moreno anyway. The Swansea defender is tried and tested in the Premier League, while the 22-year-old Spaniard is not.
A 21-year-old international left-back for around £10 million? What's not to like?
The deal makes sense from all perspectives and could help bring extra balance and options to Rodgers' Liverpool side.