James, 39, is a free agent after being released by Portsmouth last month and has been linked with moves to Chelsea, West Ham, Sunderland and Celtic is recent weeks but could now be set for a sensational switch to Steve Coppell's ambitious Championship team.
The former Liverpool, Aston Villa and West Ham keeper was due to meet Hoops boss Neil Lennon for talks at Parkhead on Monday only to pull the plug at the last minute due to family reasons.
And it has now emerged that the Robins have approached James' agent with the offer of a 12-month contract after learning of James' desire to play his football close to his home near Exeter.
"We have been in touch with David James' agent and offered him a one-year deal," a source close to the close to the club said on Wednesday. "When we heard he had pulled out of talks with Celtic, we decided to make a move last night (Tuesday).
"We have heard that he wants to remain in the West Country and we are waiting for his agent to come back to us with an answer.
"David James is a free agent and is sure to have a lot of offers but, if location is more important to him than money and status, then we feel we are in with a chance."
City are sure to face fierce competition from clubs who can offer a lot more money for the former Anfield, Villa Park and Upton Park custodian, but manager Coppell believes he can tempt James to commit to the Robins for this season.
James has racked up 536 appearances in the Premier League (including 215 for Liverpool) - a record for any keeper at that level - and played in three of England’s four games at the recent World Cup in South Africa.
And while the likes of Wayne Rooney, Steven Gerrard and John Terry were accused on underperforming, James was generally considered to have had a steady tournament.
By James Martini
