The Irn-Bru Third Division outfit, who sacked Gordon Wylde last week, have approached the 32-year-old former Wales international.
Shire owner Spencer Fearn said: "John is really keen and we had a great chat after he worked for Setanta on their game.
"He's going back down to Swansea and we'll know on Tuesday whether or not he's taken the job.
"This could be a big move for us and his first step into management."
East Stirling sit second bottom of the table as they seek to avoid a sixth successive bottom-place finish.
Hartson announced his retirement last month after being released by Coca-Cola Championship West Brom.