Rodgers has asked to leave Swansea so he can become Kenny Dalglish's successor, meaning Jenkins and the board face a tricky search to find a replacement. Swansea's attractive possession style of play won plenty of plaudits in their maiden Premier League campaign, and the first-team squad has been put together with that in mind.
"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," Jenkins said.
"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."
Swansea club captain Garry Monk admitted he was "gutted" by the news, but has backed the board to bring in the right man as the club prepare for next season.
"He's a top, top manager and we are very disappointed to lose him," he told Sky Sports News. "He's honest, treats us like adults and knows his football. His philosophy, views and training are inventive. We never do the same session more than twice and that makes it exciting.
"I think every manager is different but I think he's got the ingredients to be a top, top manager. As much as we are gutted to lose him, he's earned the right to work for a club like Liverpool. He deserves this chance.
"There will be many names linked with the job but it's important for the board to pick the right man. We're a close knit club. Everyone sticks together so we'll have to do that again."