Swansea City have rejected an improved offer of £40m from Everton for midfielder Gylfi Sigurdsson.
Leicester City also had a £40m bid for the player rejected with the club having set a £50m asking price for him.
The player will not go to the United States for Swans’ pre-season tour, heightening speculation he is to leave.
“Gylfi did not feel in the right frame of mind to travel due to the current uncertainty over his future,” said the Swans on Twitter.
Swansea have also rejected a £40m bid from Leicester City for the player, but Clement said the “business side to football” meant the club might have to sell him.
Everton have previously been linked with the former Tottenham Hotspur player.
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Sigurdsson has three years left on his contract but, having sold Jack Cork to Burnley, Clement says he cannot guarantee the Iceland playmaker will remain at the Liberty Stadium.
“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 came in with a really good offer.
“That might end up being the case with Gylfi. We don’t know yet.”
Sigurdsson captained Swansea in the second half of their 1-0 friendly defeat at Barnet on Wednesday.
Goalkeeper Erwin Mulder and midfielder Roque Mesa made their Swans debuts as their League Two opponents won thanks to a first-half goal from John Akinde.
But all eyes were on Sigurdsson, whose future Clement wants to be resolved as soon as possible.
“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,” the former Bayern Munich assistant manager added.
“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 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.”
