The Liverpool and England legend wants to get back into management after leaving Aston Villa and hopes his success at Rangers still makes him attractive.
The Foxes are looking for a boss to lead them out of the Championship at the first attempt and raise morale after the depressing end to the season.
Gerrard’s camp are close to Leicester and he will have a strong chance.
Graham Potter was their first choice when Brendan Rodgers left but he did not want to jump back into work after leaving Chelsea — though he has been sounded out again.
The former Brighton boss has also been linked with a move to Ligue 1 side Nice, which prospective Manchester United owner Sir Jim Ratcliffe owns.
Gerrard would also come with Foxes legend Gary McAllister on his backroom staff, which would appeal to fans.
Leicester had considered Russell Martin for the post but relegated Southampton moved for him first and decisively.
Gerrard has been linked with several jobs, including Leeds, and is even a possible future member of the England backroom staff.
But the Leicester job may have the most appeal.
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Leeds are on the hunt for a new boss after Sam Allardyce left after failing to keep them up in his four games in charge.
His 25-day stint means he now holds the record for shortest time in charge of a Premier League club.
Allardyce would have received a whopping £3million if he had kept Leeds in the top-flight.