The Turkish side offered a reported 600,000 for the versatile Scotland international, who would team up with former Rangers striker Kenny Miller at Bursaspor, but Ibrox manager Ally McCoist is keen to keep his current squad intact.
McCoist said: "I don't even want to know what the new bid was. I am trying to build a squad here, not dismantle one.
"Steven is a crucial part of that and I'm confident he will stay and we can clinch a new deal for him.
"Steven was superb for us last season. I have spoken with him and I feel his heart is here at Rangers."
McCoist has been on a seven-game scouting mission to South America.
"I and my staff have been recommended a few players from Argentina and this was a good opportunity to go out and have a look at some of them," he explained.
"I think I have seen just about all the top teams in the country - the likes of Estudiantes, River Plate, Argentinos Juniors, River Plate and Velez Sarsfield.
"It has certainly given me the chance to get a better handle on things in Argentina and it was a very worthwhile exercise, but I wouldn't say there is anything specific at this stage.
"There have been one or two players who have caught my eye and I might try to follow up a couple of things when I get back."