The 39-year-old was offered the job on Wednesday and his south Wales club have reluctantly allowed him to leave even though he only signed a new three-and-a-half-year deal in February.
However, the safety net of that contract means Swansea are likely to receive a pay-off of between £4million-£5million once terms are agreed on Thursday morning. Rodgers is expected to sign a three-year contract and be presented as the successor to Kenny Dalglish on Friday.
Swansea chairman Huw Jenkins admitted he could not stand in the Northern Irishman's way, although he will insist on driving a hard financial bargain with the Reds.
"Following on from discussions with Liverpool's owners, Brendan has informed us that he would like to take up their offer to manage Liverpool," said Jenkins in a statement.
"At the moment we are currently in talks with the owners to agree compensation. We are trying to finalise that within the next 24 hours."
Swansea were contacted by Liverpool on Tuesday night and Rodgers expressed his wish to speak to the Merseysiders. Having done so, he told Jenkins he wished to take up their offer.
Rodgers moved ahead of Wigan manager Roberto Martinez, who held talks with Reds' principal owner John Henry in Miami last week, in the pecking order earlier this week.
He had initially rejected their advances two weeks ago as he did not want to become involved in a wide-ranging selection process to canvass the best available talent.
However, when it became clear FSG had narrowed down their options to two he relented and discussions were conducted swiftly.