Gareth Bale has given a strong indication that he expects to remain with Real Madrid next season.
Bournemouth have signed Artur Boruc on a permanent deal following his release by Southampton.
Barcelona and big-spending Monaco have both been credited with an interest in Kompany as they look to bolster their defensive options.
However, Kompany has moved to play down the rumours and insists he is staying at City as he has "unfinished business" at the Etihad Stadium.
"Speculations and questions about next season.. I'm not going anywhere," Kompany said on Twitter.
"I love playing for City and I've still got some unfinished business."