Sigurdsson has been strongly linked with a move to Everton during this summer's transfer window, and it is understood that a deal to take the Iceland international to Goodison Park is close to being completed.
Swansea have been linked with a move for their former midfielder Joe Allen, who currently represents Stoke City, in recent days, but Clement has refused to discuss possible incomings until Sigurdsson's future has been addressed.
"I'm not going to speak about incoming players. The only situation that's important at the moment is the Gylfi Sigurdsson transfer situation. If that does go through then we can start acting on potential targets. I want to do that kind of stuff in private until there's something certain," Clement told reporters.
Sigurdsson was not involved in Swansea's goalless draw with Southampton in the Premier League on Saturday afternoon.