Rangers have confirmed to BBC Scotland that they have made a further offer to versatile defender Steven Whittaker.
On Wednesday Whittaker, 27, turned down what his agent had previously described as "a substantially improved deal".
Ken Anderson said the most recent offer was "getting closer" to what they are looking for and that he and Whittaker would soon give Rangers their decision.
Meanwhile, Bursaspor have had their fourth offer for the Scotland international rejected by Rangers.
Their third bid of 750,000 euros had earlier been knocked back by the Scottish Premier League champions.
"He'll see out the final year of his contract and we'll talk to other teams in January if Rangers keep rejecting offers," Whittaker's agent said earlier this week.
"It's up to Rangers if they want to miss out on a transfer fee."
Anderson is looking for Rangers' offer to reflect Whittaker's importance to the club.
"They say they consider him to be one of their four main players," he said.
Ibrox manager Ally McCoist has already expressed his desire to keep Whittaker.
Rangers are keen to tie the defender, who can also play in midfield, to a new long-term deal.
The club's owner, Craig Whyte, said on Thursday: "It was a derisory bid from Bursaspor and Steven Whittaker will be at Rangers next season.
"I have said no Rangers players are for sale and I expect them to be at Ibrox next season."
However, Bursaspor's public interest has alerted clubs in the Premier League in England, who have asked to be kept abreast of his situation.
The Turkish outfit signed Kenny Miller from Rangers for about £400,000 in January with the Scotland striker's contract with the club due to expire in the summer.