The 31-year-old Gordon left Sunderland in 2012 and has been troubled by knee injuries since but he has trained with Rangers and Hearts this year and appears poised to return to action next season.
And he is reportedly set to undergo a medical with Celtic, having been training with the Scottish champions recently.
Gordon began his career with Hearts and made over 100 appearances for the Edinburgh club before Sunderland paid £9m for him in 2007.
He made his debut for Scotland in 2004 and went on to earn 40 caps for his country, the most recent coming in 2010.
Were Gordon to sign Celtic, it would prompt further speculation about the future of the highly-rated Fraser Forster, who is with the England World Cup squad at the moment and has been linked with a move back to the Premier League this summer.