Sigurdsson is yet to sign a contract with the Swans despite agreement on a club-record £6.8 million deal to sign him permanently from Hoffenheim following a successful loan spell.
Rodgers had pushed the Welsh club to sign Sigurdsson, who he also worked with at Reading, but the Northern Irishman's move to Anfield could prompt a rethink by the player.
"Gylfi came to Swansea because I was there as manager and his scoring record was fantastic," Rodgers said.
He has vowed to let Sigurdsson talk to the new Swansea boss first - after agreeing with chairman Huw Jenkins not to poach any of his former players for 12 months - but said he would make a move for the Iceland international should any talks break down.
Rodgers added: "I pushed the club to make an agreement [to sign Sigurdsson permanently] and they did that in terms of the fee - and agreed the salary.
"So I have said to Huw: 'I'm not going to do anything untoward.
"This is a guy you want, but it is down to Gylfi'.
"If he comes on the market I have to be interested. He knows that I rate his qualities and abilities.
"I said to Huw: 'If I speak to the kid I'll tell him to speak to the new manager, whoever that is'.
"But if he's not comfortable with that, he's in the market place then - and I have to look at him."