The Northern Irishman has agreed a three-year deal as Reds boss and will take assistant manager Colin Pascoe, performance analyst Chris Davies, and head of sports science and medicine Glen Driscoll with him.
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The Swans broke the news on Thursday evening through the club's official website and the appointments are expected to be confirmed at a press conference called for 10:00 at Anfield on Friday.
Rodgers emerged as a late candidate to succeed Kenny Dalglish at Anfield after initially rebuffing an approach from owners Fenway Sports Group.
The 39-year-old was said to be unhappy with the comprehensive list of managers in which the Merseyside outfit had expressed an interest.
However, following the reported withdrawal of Wigan boss Roberto Martinez, Rodgers expressed his wish to enter talks and has now accepted the role.
The former Reading man led the Welsh side to a 10th-place finish in their maiden Premier League season, following promotion via the Championship play-offs.
Meanwhile, the Reds slumped to a disappointing eighth which, despite League Cup success and an FA Cup final appearance, was enough to see Dalglish's time at the club concluded.