The England midfielder is in the final year of his £100,000 a week contract at Manchester City, and is reportedly concerned he is not in the plans of new City manager Manuel Pellegrini.
Barry has been at the club since the summer of 2009, and found himself a first team regular under both Mark Hughes and Roberto Mancini.
But with City planning a squad overhaul this summer after such a disappointing campaign last time around, it appears Barry may find himself surplus to requirements at the Ethihad.
City are understood to have offered him a new contract but on reduced terms, meaning he is unlikely to be considered a first team regular next season, and with the 2014 World Cup looming, the former Aston Villa man is seeking a move.
Arsenal appear in pole position to sign the 32-year-old, with Wenger keen to add some Premier League experience to his midfield.
The Gunners are preparing a 12-month deal worth £75,000 a year with the option of a further year should Barry prove a success.
London rivals Tottenham and West Ham are reportedly also interested in the midfielder.