Osman, a former teammate of Rangers number two David Weir, is said to be top of Warburton's list of Ibrox targets.
Warburton is said to be keen on adding experience to his Ibrox midfield, having lost both Niko Kranjcar (injury) and Joey Barton (contract termination) in recent months.
And Osman, who made 352 Premier League appearances in 16 seasons at Everton, would certainly fit the bill.
Reported Rangers target Leon Osman in action for Everton
The 35-year-old was released by the Toffees in the summer and could have still signed for Rangers, or any other club, between the two transfer windows.
Rangers, whose assistant David Weir played alongside Osman at Goodison Park, are also said to be stepping up their interest in Barnsley captain Conor Hourihane - another central midfielder.
Barnsley's Conor Hourihane (right) is also said to feature highly on Warburton's shortlist
The Irishman is out of contract next summer and recently rejected the Tykes' offer of a new deal.
Brentford playmaker Jota was previously described by the Scottish Sun as being Warburton's top target, and is said to remain on the 54-year-old's list along with Sergi Canos of Norwich City.
Would Osman be a good signing for Glasgow Rangers?