The 27-year-old played a pivotal role in keeping the Swans in the top flight last season, scoring nine goals and creating 13 others, and has been linked with a move to Everton.
Swansea slapped a £50million ($64 million, 56.4 million euros) price tag on him, with the Welsh club having rejected a £40m bid from 2015-16 champions Leicester.
According to the Press Association, Everton had a bid for a similar amount rejected by Swansea on Thursday.
“You don’t want to be in a situation where it’s dragging on through July and all through August, when the season’s started, with the situation not being resolved. The sooner the better,” said Clement prior to the squad leaving for their pre-season tour in the United States.
“I don’t know how people are going to behave in the future. I know the kind of character I worked with last year — someone who’s very professional.
“He’s (Sigurdsson) come back, he’s got his head down and he’s doing his work, so I hope that continues.
“He’s one of our most valuable players. He showed that last season with his goals and his assists, and he was instrumental in us maintaining our Premier League status.”
However, Swansea tweeted later on Thursday that Sigurdsson would not be travelling with them on their pre-season tour.
“Having played v Barnet last night, Gylfi did not feel in the right frame of mind to travel due to the current uncertainty over his future,” the club said on Twitter.
Clement, who admitted on Wednesday that Spanish striker Fernando Llorente may not be fit for the start of the campaign due to a broken arm, said he hoped Sigurdsson would stay.
However, he admitted that if the bid was substantial enough business would dictate he left — although it was also reported Swansea declined an offer from Everton of money plus their England midfielder Ross Barkley.
“I’ve made it very clear I want him to stay here,” said Clement.
“The fans want that, the owners would like that but, ultimately, there’s a business side to football as well.
“We had that situation with Jack Cork. In the summer there was no thought of moving Jack on but it got to a point when another club (Burnley offered £8m with add-ons) came in with a really good offer.
“That might end up being the case with Gylfi. We don’t know yet.”
Sigurdsson, whose contract runs to 2020, has been at Swansea on a permanent basis since 2014 after a disappointing two-year spell at Tottenham, who bought him from German outfit Hoffenheim for nearly £9m.
Despite Koeman spending a lot of money already in the close season, the £75million due from the transfer of Belgian international striker Romelu Lukaku to Manchester United should give him sufficient funds.