Clarke has transformed Killie since taking the reins in October, with the club securing a top-six spot despite fears early on in the campaign that they would be embroiled in another relegation battle.
Clarke’s work with Kilmarnock saw him tipped for the Scotland job, but there are now murmurings about the former West Brom boss being high up on Rangers’ list of candidates should Graeme Murty leave.
But Bowie is having none of it.
He told the Daily Record: “Let’s nip this in the bud. There’s been a lot of speculation, but Steve signed a two-and-a-half year deal and as far as we’re concerned he will be here next season.
“There seems to be all this talk as if he’s definitely going here, there and everywhere in the summer as if it’s a done deal. Far from it. We heard it before and this week it’s Rangers. No one really mentions little old Kilmarnock.
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“I understand people might be looking at the job he is doing, but we are trying to build something here and Steve is a huge part of it. He has never once mentioned anything about going anywhere and is completely focused on his job.”
Kilmarnock secured a place in the top six with victory over Murty’s Rangers at Ibrox, combined with Motherwell failing to beat Celtic on the Sunday.
Now Bowie, who has hailed Killie fans for returning in their numbers to back the team, believes Kilmarnock are on the cusp of something special.
He continued: “We knew we were bringing in a top manager but I don’t think anyone expected the kind of impact he has made.
“There were 9,000 at Rugby Park last week, on a cold Tuesday night (for the Scottish Cup replay against Aberdeen). The result was disappointing but it shows how far the club has come under Steve.
“To secure a top six place so soon is a terrific achievement. The season is far from finished yet and the next target is fourth.
“Who knows, we could even get a couple of trips to Europe next season, which would be incredible. But it’s not just about what the club does this season.
“There are plans in place for the future and Steve has a big role to play.”
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