Sigurdsson is yet to sign a contract with the Swans despite agreement on a club-record £6.8million 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.
Rodgers 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.
"Gylfi came to Swansea because I was there as manager and his scoring record was fantastic," he told various Sunday newspapers.
"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'."