FSG were also known to be keen to speak to Rodgers, but a Swansea statement read: "Swansea City can confirm that an approach from Liverpool FC to speak to manager Brendan Rodgers was officially received by chairman Huw Jenkins on Friday morning.
"The club is pleased to confirm that Brendan has declined the current opportunity to speak to the Anfield club about the vacant position."
The statement continued: "The Swansea manager is currently working hard to strengthen the squad in readiness for the new Barclays Premier League campaign.
"The club will make no further comment at this stage."
Rodgers, who signed a new three-and-half-year deal in February, has previously been linked with Chelsea and Tottenham after making a huge impression by guiding Swansea to an 11th-place finish in their maiden Premier League campaign.
The Welsh club rounded off their season by beating Liverpool 1-0 at the Liberty Stadium, a fixture which proved to be Dalglish’s last in charge of the Reds.
Swansea’s attractive possession style of play also won former Reading and Watford manager Rodgers plenty of admirers, and chairman Huw Jenkins has admitted the Swans will have to become use to their boss being linked with other clubs.
And it says much about the Northern Irishman’s burgeoning reputation that he was among the bookmakers’ favourites to take over at Anfield, along with the likes of former Swans’ boss Martinez, Andre Villas-Boas, Frank Rijkaard and Didier Deschamps.
But the decision to reject an approach from one of the British game’s traditional powers represents a major boost to the Welsh side as they look to build on a promising start to life in the top flight.
Despite the speculation linking him with other clubs, Rodgers has always insisted he is committed to Swansea.
After being linked with Tottenham in February, he said: "The most important thing for me is to have peace and satisfaction in my job, I don’t know where else other than Swansea that I can be as happy as I am.
"If speculation comes with me doing the job well then I am flattered and honoured. However, this is the place I want to be, hopefully I can be here for the duration of my contract, that’s my aim.
"I am already at a top club, Swansea is a top club. It might not be in the Champions League and, at this moment in time, challenging for trophies, but in my own mind I am at a top club.
"This is a club that is run fantastically well, I get the opportunity to manage this club how I want to and do everything here.
"I am 39 years of age. I hope to prove myself over a long period and my only concentration is on Swansea City."
Rodgers is due to fly out to Austria this weekend as he gets the chance to spend four days with world and European champions Spain at their pre-Euro 2012 training camp in Schruns, before heading to New York to take in Wales’ friendly with Mexico.
Rodgers is also to hold talks with Tottenham over taking defender Steven Caulker on another season-long loan, while Swansea are thought to be edging closer to agreeing a permanent move for Gylfi Sigurdsson after the Hoffenheim midfielder’s stunning loan spell.