The France international will see his current contract with the Gunners expire at the end of the 2011/12 campaign.
Arsenal have expressed a desire to tie him down long term, but the 24-year-old midfielder is reluctant to pen fresh terms.
A lack of silverware at the Emirates has forced Nasri to consider his options, and he believes a switch elsewhere may be beneficial to his career.
Manchester giants United and City are reported to be keen, with the latter prepared to push through a big-money move during the summer transfer window.
Backing
Arsenal, though, are refusing to give up hope in their efforts to thrash out a new contract with Nasri and admit they are willing to let the former Marseille man run down his deal.
Chairman Peter Hill- Wood said: "It is possible we could lose the player on a free transfer but we are prepared to take that risk."
The Arsenal board have also vowed to back Arsene Wenger in his bid to keep other key men in north London.
Reports continue to suggest that club captain Cesc Fabregas could soon be returning to his Catalan roots with Barcelona.