Rodgers hopes that Sigurdsson's love for the club and the city will be enough to tempt him to stay in Wales permanently.
"I think everybody has seen his abilities and qualities since he's come in," he said. "A goalscoring midfield player is very hard to find but that will be up to the board and the club to see what they can do [about signing him].
"Gylfi loves being here. He loves his training. He loves the work. He loves the city. So we'll see how that goes between now and the end of the season."