Steven Naismith (left) scored for Kilmarnock against Dundee

Kilmarnock manager Jim Jefferies told BBC Sport that their Scottish Premier League rivals had not been in official touch for a couple of weeks.

And Ibrox counterpart Walter Smith said: "We have withdrawn the offer. I can't make it clearer than that.

"We can't come to agreement with Kilmarnock, so that's it finished."

The transfer of Naismith appeared near to completion last week when Jefferies revealed that informal talks through the player's agent had brought the two sides closer together.

But Killie chairman Michael Johnston told BBC Sport: "It hasn't resulted in anything concrete.

I hope Kilmarnock accept an offer soon and let me prepare properly for the new season Kilmarnock striker Steven Naismith

"Rangers have improved the terms under which payment would be made, but there is still a fair gulf between our valuations.

"Meantime, we have had a higher offer from another major club, but nobody has offered what we are looking for."

Reports suggest that Kilmarnock want £2m, while Rangers' latest offer was closer to £1.5m.

Celtic, Bristol City and Hibernian have previously been linked with the player.

Naismith scored Killie's second goal in Saturday's 2-2 draw away to Dundee in what was his first pre-season outing.

And Jefferies admitted that his movements in the transfer market were being constrained by the uncertainty surrounding the 20-year-old.

MY SPORT: DEBATE The only other "major" club it could be is Celtic Killie Boy 118

"Rangers have not been in touch officially for a couple of weeks," he said.

"There has been interest from other parties, but nothing concrete, so there is nothing new to report."

Meanwhile, Naismith showed his frustration about the lack of progress.

"I just want my future sorted out as quickly as possible," he told the Sunday Mail.

"All the speculation is starting to affect me.

"Whatever club comes in for me, I hope Kilmarnock accept an offer soon and let me prepare properly for the new season.

"This could be a life-changing opportunity and I don't want anything to jeopardise it."