Naismith won his first senior cap for Scotland last week

Rangers have had a bid for the 20-year-old rejected, while Celtic are also monitoring the situation.

And the player, who has three years of his Kilmarnock deal to run, has ruled out a move to England.

"We might be prepared to improve his contract without extending it to keep him happy," said Johnston.

The Ayrshire club recently rejected a bid from Rangers of �400,000 for the highly-rated Scotland international.

And Johnston will not sell Naismith unless the club receive a vastly improved offer.

"As I've said before, we are keen to keep him," added the chairman.

"Steven is improving all the time and he knows he is guaranteed first-team football at Kilmarnock, which he might not get elsewhere.

"His progress will continue here if he stays and I'm sure he will go on to win many more Scotland caps.

"We are not inviting bids and we have not asked for them."

Naismith's representative Andrew McCormick told the Scottish Sun: "There hasn't been an offer from Celtic so far but there is definite interest.

"Steven does not want to go down south at this stage and that only leaves two clubs to choose from.

"People think he would choose Rangers over Celtic because he supported them as a boy but that's not the case.

MY SPORT: DEBATE Where do you think Naismith will end up?

"Those days have gone and, if both make offers, then he would have a tough decision on his hands.

"If it comes down to Rangers and Celtic and they make matching offers, it will be a footballing decision."

McCormick has made it clear that his client wants to remain in Scotland for family reasons.