The Swans agreed a fee worth £6.8 million with Hoffenheim last month to take the Icelandic midfielder to the Liberty Stadium permanently following a successful loan spell.
Personal terms are believed to have been agreed between the club and the 22-year-old, but Rodgers’ move to Anfield has thrown the deal into doubt.
Sigurdsson has been tipped to follow the 39-year-old to Merseyside, after claiming he is unlikely to sign for the Premier League side following Rodgers’ departure.
However, the new Reds boss, while admitting he is keen on the goal-scoring midfielder, insisted his former club must be given the chance to convince Sigurdsson to remain in Wales.
"First and foremost, Gylfi is a Hoffenheim player," he told Liverpool’s official website.
"I took him on loan for the second half of last season and he did very, very well. He scored seven goals and made a real, positive contribution.
"I wanted to sign him for Swansea but, like I've said to him and his representatives, that has to be his first port of call.
"If he comes into the market for whatever reason - and I'm sure there'll be a number of clubs interested in Gylfi - then of course I would like to be in a position to put our case here at Liverpool to sign him.
"But I've always said, I think he's a player Swansea have had an agreement with and they have to have every chance to try and sign him. That's only right.
"If anything changes from that, we'll look at it from there."
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