Mark Warburton, who is keen to strengthen his defensive options, has targeted the experienced Aston Villa man.
It is understood the Gers boss met the 34-year-old today in a bid to lure him to Ibrox.
Lescott was relegated from the Premier League last season and still has a year left on his Villa deal.
But new Villa manager Roberto Di Matteo has signed two centre-halves – Tommy Elphick and James Chester – with Lescott told he can move on for free if he can find another club.
Former Arsenal defender Philippe Senderos, 31, has been training with the Gers since Monday as he attempts to win a deal. But Lescott appears to have moved ahead of him in Warburton’s signing plans.