Bony has been linked with the likes of Tottenham, Everton and Liverpool during the close-season, while there have also been rumours the Ivory Coast international will join former manager Michael Laudrup at Lekhwiya in Qatar.
To add to the speculation the 25-year-old might leave, Swansea have signed Bafetimbi Gomis for the next four seasons with the former Lyon forward a very similar player to Bony.
But with Bony having scored 17 goals in 34 Premier League games for Swansea last season and three years remaining on his contract, Monk made it clear on Monday that he does not want the powerful striker to depart the Liberty Stadium.
"There will be interest in a number of our players - that's the way it is - but at the end of the day, Wilf is a Swansea City player," Monk told the South Wales Evening Post.
"Everything that has been said about him this summer has been said in the papers. We are not a paper company, so we'll just deal with the reality.
"At the moment nobody has come in for Wilf and he is our player.
"But it can't drag on for the whole summer.
"We don't want one of those scenarios where it's going on right up until the last day.
"It has to be made loud and clear what is happening at some point - but at the end of the day he is not for sale unless someone comes in with an astronomical offer."
Monk insisted Bony is not angling for a move either.
"He knows he is Swansea City player and he knows nothing has happened so far," the 35-year-old manager said.
"All he wants is to concentrate on what he is doing for Swansea City.
"That's why he is with the squad [in Devon]. If he was thinking about going, he wouldn't be here."
Swansea reportedly paid £12million for Bony in 2013 to bring him to Wales from Vitesse Arnhem.
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