Galatasaray chairman Unal Aysal told Turkish newspaper Milliyet that United rejected their approach for the winger, who is out of contract next summer.
And Aysal revealed the Premier League champions are holding out for closer to £8.5m.
Speaking to Milliyet, Aysal said: "We offered €7m (£6m) for him. But they didn't accept. We can't give more than this.
"We are not going to follow Nani anymore. We can't give more than €7m for him. They want €10m (£8.5m). Because of this he is not on our transfer list anymore."
Nani has been strongly linked with a move away from Old Trafford this summer, with Ligue 2 champions Monaco reportedly interested.
He endured a disappointing campaign last season, making just seven Premier League starts. And the 26-year-old recently admitted his future could lie away from United.
But it seems the Old Trafford club are holding out for a bigger fee for the player that joined from Sporting Lisbon in 2007.
Galatasaray have been big players in the transfer market over the past six months, bringing in Didier Drogba and Wesley Sneijder. And the Turkish champions made a run to the Champions League quarter-finals only to fall to Real Madrid.
Nani's place in the United first-team squad could come under fresh threat next season when £15m signing Wilfried Zaha joins the club this summer.