The 39-year-old Northern Irishman rejected an initial approach from Fenway Sports Group (FSG) due to the large number of candidates reportedly being considered for the role.
But FSG, who had already held talks with Wigan boss Roberto Martinez, made a renewed approach to Swansea chairman Huw Jenkins on Tuesday night, with Rodgers on this occasion keen to hold discussions with Anfield chiefs.
Those talks were held on Wednesday afternoon and Rodgers has asked to leave so he can become Kenny Dalglish's successor, with the two clubs now holding discussions over compensation.
Jenkins said in a statement: "I was contacted by Liverpool and they expressed their wish to speak to Brendan regarding their vacancy.
"I had a discussion with Brendan to talk about their interest and his views on whether he wanted to speak to Liverpool.
"He expressed his wish with me to do that and he has spoken to Liverpool.
"Following on from discussions with Liverpool's owners Brendan has informed us that he would like to take up their offer to manage Liverpool.
"At the moment we are currently in talks with the owners to agree compensation.
"We hope to finalise that within the next 24 hours."