"Following on from discussions with Liverpool's owners, Brendan has informed us that he would like to take up their offer to manage Liverpool," Swansea chairman Huw Jenkins said on the club's website on Wednesday.
Swansea said discussions were still ongoing with Liverpool's American owners over compensation for Northern Irishman Rodgers, who guided the Welsh side to a respectable 11th-place finish in the Premier League last season, their first campaign at the highest level since 1983.
"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," added Jenkins.
"Although we are very disappointed to lose such a talented, young British manager, we didn't wish to stand in his way. We will always remain good friends and we thank him for his hard work and passion at this football club over the past two years."
Liverpool have been searching for a replacement for Anfield great Dalglish, whose second spell in charge ended after he was dismissed following the club's eighth place finish in the Premier League, their worst for almost 20 years.
Once the dominant force in English football, Liverpool have not won the championship since 1990. They finished their 2011/12 campaign 37 points behind champions Manchester City and recorded their fewest wins in a top-flight season since 1953/54.
Wigan Athletic manager Roberto Martinez also held talks with Liverpool but Wigan chairman Dave Whelan told Sky Sports News that the Spaniard would remain in charge.
"He moves into the second year of the deal previously agreed," Whelan said.
Rodgers, 39, and Swansea won plaudits for their eye-pleasing passing game and their 1-0 victory over toothless Liverpool in their final match of the season could not have failed to impress the Anfield hierarchy.
Whether Liverpool's long suffering fans will be convinced remains to be seen with many unhappy at Dalglish's departure after a League Cup triumph and FA Cup final appearance.
After spells in charge of Watford and Reading, Rodgers moved to Swansea in 2010 and guided the Welsh club to promotion from the Championship via the play-offs.
Dutchman Louis van Gaal has been linked in the media with becoming Liverpool's new Director of Football after Damien Comolli left in April.
