Swansea are yet to confirm or deny if they have accepted Everton's offer but, considering they also rejected a £40m bid from Leicester for the Reykjavik-born footballer, it would be odd if they accepted an identical offer from Koeman's team.
Sigurdsson played in a pre-season friendly at Barnet on Wednesday evening, but a City statement the following morning confirmed he would not be travelling to the US.
The player has been heavily linked with Everton and Leicester City, and both clubs have reportedly had a £40m bid turned down by Swansea, who are holding out for £50m.
Manager Paul Clement has previously insisted Sigurdsson will only leave the club on their terms: 'He is training hard and straightaway he is at the top of all the statistics physically, ' said the Swansea boss.More news: The GKN plc (LON:GKN) Stock Rating Reaffirmed by Deutsche Bank AG
"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".
Paul Clement, who is desperate to keep the 27-year-old in order to help improve their league footing, has admitted he does not know what the future lies for his main man. I know the kind of character I worked with past year - someone who's very professional. The fans want that, the owners would like that but, ultimately, there's a business side to football as well.
"He's one of our most valuable players".
He went on to say: "That might end up being the case with Gylfi". We are not sure whether he is going to be here for this season.