The 25-year-old will undergo a medical on Monday before signing for an initial £500,000. Davies' deal also includes structured add-ons - primarily related to first-team appearances and any achievements the player contributes to - worth up to £1.1million.
Preston will also benefit from a 20 per cent sell-on clause on any profit Liverpool make on any further transfer and also any England appearance. North End will also get the 19-year-old defender Sepp Van Den Berg on loan for free with the first option for a loan next season should he start a certain percentage of matches in the current campaign.
Davies, a left-footed centre-back, is in the final six months of his contract and it was expected he would sign a pre-agreement with Celtic to join them in the summer.
However, it emerged the reigning Scottish champions, currently 23 points behind Rangers at the top of the table, were planning to make a deadline-day offer to Preston and that prompted Liverpool to make their move.
Fellow Premier League sides Burnley and Sheffield United were also interested but Preston did not want to sell to their local rivals but it is understood the player was not interested in either club due to their style of football.
Celtic are now turning their attention to other targets and have been linked with a deadline day swoop for Tottenham Hotspur centre-back Japhet Tanganga on loan for the rest of the season.