The Hoops were initially aiming to land the 25-year-old on a pre-contract agreement with his Preston contract expiring in June.
However, they are now keen on signing him this month, provided they are able to successfully negotiate on the personal terms.
Championship side Bournemouth are also interested in Davies. Preston want around £3 million from the south coast outfit.
On the other hand, it is reported that the Hoops could secure the player’s services for a lower fee between £2m and 2.5m during the current transfer window.
Davies has been a key player for Preston in central defence, starting 18 games in the Championship this term.
Despite this, they appear open to parting ways with the defender, who can also play at left-back, to avoid losing him for nothing in the summer.
They are in a strong position to sign Davies, who has turned out 144 times for Preston since graduating through their academy.
It remains to be seen whether Bournemouth are prepared to meet the £3m price tag and convince the player to join them instead.
It is quite obvious that Preston don’t want to strengthen a direct promotion rival and are therefore demanding a higher sum from the Cherries.
Stats from Transfermarkt.com