The Swans offloaded the Iceland international for a reported fee of £45m on Wednesday evening and now have just two weeks left to bring in replacements.
Clement has made clear that he intends to sign "two or three" more players before the window closes for business, but he is wary that teams will now hold out for additional money for certain targets.
Asked specifically about speculation linking the Welsh outfit with Wilfried Bony and Nacer Chadli, he told reporters: "The emails and the texts are coming in from agents all over the world. Because they know we've done the Gylfi Sigurdsson deal and they know that we want to bring players in.
"But we're focused and what our needs and our priorities are. We have multiple targets because sometimes things don't come off or the price is too high.
"What's important now is we move swiftly because time is important but we certainly won't panic. Sometimes the deals are better at the end of the window.
"That can sometimes happen [when clubs ask for higher fees] but we're going to make sure we pay the right valuation for any player we buy. Our net is cast wide and the way the market is now, especially with British players, sometimes you have to start looking elsewhere."