Sigurdsson impressed while on loan from Hoffenheim at the Liberty Stadium in the second half of last season under former Swans boss Rodgers.
The Welsh side agreed a club-record fee of £6.8million for the 22-year-old Iceland international last month and confirmed he had accepted personal terms.
However, Rodgers' move to Anfield has thrown the potential deal into doubt, with Sigurdsson tipped to follow him to Merseyside.
The youngster revealed earlier this month he is now unlikely to sign at Swansea following the manager's departure.
But Rodgers, while still keen on Sigurdsson, insists his former club must be given the chance to persuade the midfielder to stay in South Wales.
He told Liverpool's official website: "First and foremost, Gylfi is a Hoffenheim player.
"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."
Rodgers says he has several targets in mind as he looks to improve the Liverpool squad this summer but, other than Sigurdsson, the manager is keeping his cards close to his chest.
"I like to very much keep my counsel on potential targets," he said. "I think there are a few players who can come in and improve the group straight away.
"I know the types of player I need to bring here to make us function well and that's something myself, the club and the board will look at.
"I know what we've got, more or less, in the squad, but I also know there are a couple of signings that could really improve what we do.
"That's something we'll work together on over the course of the summer and hopefully we can get one or two in."
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