The former City right-back is a free agent after his contract was allowed to expire at the Etihad Stadium earlier this summer, with moves to both Crystal Palace and Stoke City failing to materialise.
The 34-year-old would prefer to continue playing in the English top flight but his previous £90,000 per week City wages are believed to have scared away a number of interested suitors.
Italian side Torino expressed a recent interest but it is the Foxes who are suggesting a potential compromise for the ex-France international, who may even be willing to accept a pay-as-you-play deal, coupled with a decent basic salary of around £50,000 per week.
It is understood Leicester boss Craig Shakespeare may be open to signing the full-back on a short-term deal for the rest of the campaign, with the promise of something more concrete if he can help the former Premier League champions solidify their leaky backline.
The one-time Arsenal man has kept himself in decent shape and is ready to answer the Foxes' call if the two camps can now make the numbers work.