Rangers have had four bids rejected for Naismith

"We have had an offer in excess of what Rangers have offered," said Rugby Park manager Jim Jefferies.

"It's a verbal offer at the moment, with some additional add-ons still to be discussed to take us up to what we're looking for.

"We expect to hear back from that club in a couple of days."

Rangers submitted the same offer of £1.5m which was dismissed by Kilmarnock at the weekend, but have structured it differently.

This time, Rangers offered £1m to be paid in two instalments and lowered the number of appearances at which a further £500,000 would be due.

MY SPORT: DEBATE Are supporters getting fed up with the Naismith transfer saga?

The 20-year-old Naismith flew to Italy with Kilmarnock on Thursday for a pre-season training camp.

And Jefferies insisted: "We have never had an agreement with Rangers. They are entitled to their valuation and so are we.

"They have tried to get him for as little amount as possible, as you would expect any club to do.

"But they have known from a few days after they made their first ridiculous offer what we are looking for.

"If Rangers want to pull out, that's up to them. Steven will be a bit disappointed but he will get on with his life at Kilmarnock."

Celtic have been credited with an interest in the SPFA Young Player of the Year and there have been reports of a surprise bid from Hibernian.

"We're not trying to sell him or get into an auction," added Jefferies.

"We have always been straight and asked for a fair price. If we agree a deal and he doesn't want to go there, he doesn't have to go.

"He has handled all of this well but it must be unsettling which is why we want things resolved as soon as possible."