The Foxes were keen for former Leeds United boss Jesse Marsch to take over, but the American was unsure and unsatisfied with answers to his questions about what might then happen in the summer.
Moving quickly, Leicester have now turned to Smith and are in advanced talks with him.
They want Smith to come in on an interim basis to guide the side through to the end of the campaign.
Smith has been out of work since being sacked by Norwich in December and is eyeing a return to management.
He struggled to live up to expectations at Carrow Road, but could now be set to return to the dugout in the Premier League.
Leicester are deep in trouble in the Premier League and were beaten by fellow strugglers Bournemouth at home at the weekend.
Smith, if he takes over, is likely to have to lead Leicester into their meeting with Manchester City next weekend in a baptism of fire.
