Rodgers turned down an initial approach from Reds' owners Fenway Sports Group two weeks ago due to concerns over the number of candidates being linked with the job, 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 and Rodgers has asked to leave so he can become Kenny Dalglish's successor, with the two clubs hoping to finalise a compensation package, set to be worth between £4million and £5million, in the next 48 hours.
The Swans may find it difficult to find a replacement for Rodgers who would be prepared to wholeheartedly embrace his philosophy, but Jenkins said: "Although we are disappointed to lose such a talented, young British manager, we didn't wish to stand in his way.
"As always at Swansea City, we want people working here who are fully committed to the task ahead. We wish Brendan every success in the future.
"We will always remain good friends and we thank him for all his hard work and passion at this football club over the past two years.
"We shall now refocus and quickly start the process of finding his replacement to continue the great work Brendan has carried out at this football club."