Milner, 29, has been one of Rodgers' long-standing targets and is available on a free transfer when his contract at Manchester City ends later this month.
One of the players the Liverpool manager considers key to his rebuilding process following the departure of captain Steven Gerrard, Milner would sign a pre-contract committing him to a long-term deal at Anfield when his City contract expires on 30 June.
Liverpool also retain an interest in the Aston Villa striker Christian Benteke but are not prepared to pay the £32.5m buyout clause in the Belgian's contract.
The Villa manager, Tim Sherwood, said in the aftermath of the club's FA Cup final defeat on Saturday that Benteke would not be permitted to leave for less than that sum, however Liverpool hope that they can persuade Villa to sell him for less.
Milner has been a highly sought-after free agent, attracting interest from Arsenal as well as Liverpool, but Rodgers is confident that he will sign for the Merseyside club. The England international joins up with Roy Hodgson's squad at St George's Park tomorrow ahead of games against the Republic of Ireland and Slovenia.
The winger is the first of two signings the club want to announce, with Danny Ings, 22, another free agent, having given his assurance that he, too, will join when his Burnley deal expires at the end of this month.
Liverpool are yet to agree a compensation deal with Ings' club Burnley but they are hopeful of coming to an agreement in the next month to avoid the case going to a transfer tribunal.
In the meantime, Rodgers will meet Tom Werner this week as the club chairman will be over from Boston to attend the Premier League shareholders' meeting on Thursday.
Werner will be joined by Mike Gordon, the second biggest shareholder in FSG, after both men arrived in Liverpool yesterday. Over the next three days they are due to meet Rodgers to review the season and also hold meetings about the development of Anfield and the club's academy.
There is no question that Rodgers' job is under threat in the short term. However, it was significant that Jürgen Klopp, a potential successor at Liverpool, ruled out any possibility that he might take a job this summer following his departure from Borussia Dortmund at the end of the season by confirming he was going to take a break.
In a statement, Klopp said: “After seven intense and emotional years, I think it's a good idea to let the numerous memories settle in before I take on a new challenge. I'm going to take a break until further notice.”
Klopp, who led Dortmund to the Champions League final in 2013, lost his final game in charge on Saturday as Wolfsburg claimed the German Cup with a 3-1 victory. – The Independent
Original source: Liverpool won’t pay £32.5m for Benteke