Naismith won the Young Player of the Year award for 2006/07

Rangers remain favourites to land the 20-year-old but have so far had four bids rejected, with a £1.5m reported price tag on the Scotland player.

But Jefferies told BBC Scotland: "In writing Rangers are a bit away but there's one or two verbals going back and forward through a third party.

"And it seems they're not far away."

Interview: Kilmarnock manager Jim Jefferies

"But there's nothing official yet.

"Whether we ever get to that stage I don't know," he added.

Jefferies said on Tuesday he had resigned himself to losing Naismith before the start of the new season.

He told the Daily Express: "I've resigned myself to the fact Stevie is going.

I've resigned myself to the fact Stevie is going Kilmarnock manager Jim Jefferies

"Whether it takes a few days or a few weeks, we don't know."

Jefferies had hoped to keep Naismith - who made his debut for Scotland against the Faroe Islands in June - and insisted he could benefit from another season at Rugby Park.

"If Rangers do agree on a fee then I think he would go there," said Jefferies.

"I would be very happy to have him here for next season, but we can't hold the boy back."

The Killie boss had earlier insisted that the Naismith signing saga was not affecting team morale.

"To suggest there's any effect on the squad is sheer and utter nonsense," Jefferies told BBC Scotland.

MY SPORT: DEBATE Give your reaction to the Naismith transfer saga here

"He's been strongly linked with Rangers but we've had no contact with them for well over a week now.

"What we've asked is a fair and reasonable price for a tremendous talent, and if any club offers that money he'll be free to talk."

The Ibrox outfit's four rejected bids have included add-on fees linked to Naismith's future achievements.

And Rangers are still in the hunt to sign the exciting forward but may be prepared to wait until August to relaunch their bid, unless Killie enter negotiations with other teams.

With the player's future still open to question, Jefferies accepts Naismith's emotions will be "in a bit of turmoil".

"As far as Steven and I are concerned, he knows that I've kept him up to date with everything," said Jefferies.

"He'll probably be in a bit of turmoil at the moment. He hears this and that and he wants it resolved.

They (Rangers) do know what we're looking for and it's a reasonable amount for a player of his quality Jim Jefferies

"They do know what we're looking for and it's a reasonable amount for a player of his quality.

"I don't think we can be accused of pricing him out of the market.

"People get a chance to go where they like when they're out of contract. They can choose their club.

"But, when they're in contract, it's always the same for every transfer - the two clubs have to agree a deal.

"The key thing to this is the club aren't asking for ridiculous money."

Jefferies also admitted that he will not receive any money for new signings from any sale of Naismith.

"It's been made very clear that the money would go towards the club," he said.

"Obviously we operate with a budget for wages and we have to stick to that.

"That money would then take a lot of the pressure off because we're still burdened with debt."