Bryson has attracted the attention of former Rugby Park boss Jim Jefferies
Kilmarnock manager Mixu Paatelainen has targeted a striker but insists he does not have to offload midfielder Craig Bryson to Hearts to acquire one.
The Ayrshire club have turned down a bid from their Scottish Premier League rivals for their 23-year-old captain.
"They were way, way below our asking price," Paatelainen told BBC Scotland.
"We are looking to strengthen the striking department as we do not score enough goals, but we hope Craig stays and I will be delighted if he does."
Hearts manager Jim Jefferies has already raided Rugby Park for Kevin Kyle this summer after the Scotland striker rejected a new contract.
And the man who signed Bryson for Kilmarnock from Clyde in 2007 is keen to take the former Scotland Under-21 international to Tynecastle, although he is also interested in signing veteran Scotland midfielder Paul Hartley from Bristol City.
There's a little money. Not too much
Kilmarnock manager Mixu Paatelainen
Paatelainen's predecessor, Jimmy Calderwood, left Rugby Park at the end of last season, putting the blame on the size of budget for players and other football staff.
The Finn knows the constraints under which he has to work at debt-hit Kilmarnock but thinks he can add to his squad without having to sell prize asset Bryson.
"A striker is our priority, but I am looking to strengthen in all departments," said Paatelainen, whose club released forwards Allan Russell and David Fernandez this summer.
"There's a little money. Not too much.
"We are quite limited in what we can do, but we will dig deep."
They have already done so to sign Guillaume Beuzelin, pending a medical, after the French midfielder was released by Hamilton Academical.
Paatelainen had no specific squad size target, pointing out that the versatility of any new arrivals could reduce the numbers required as cover.
Kilmarnock's manager hopes to run the rule over some trialists in Devon during this week's Joma 2010 South West Challenge Cup, which involves 13 clubs in four groups.
The side who avoided relegation from the Scottish Premier League in the final game of last season kick-off the tournament on Monday against hosts and Western League Champions Bideford at the Sports Ground.
Kilmarnock on Thursday face Blackpool, who are preparing for their return to England's top flight, at the same venue.
Blackpool will by then have faced League Two side Accrington Stanley, who also take on Bideford on Wednesday, with the group winners progressing to Saturday's semi-finals.
The other tournament participants are Doncaster Rovers, Shrewsbury Town, Leyton Orient, Rushden & Diamonds, Barnstaple, Truro, Eastleigh, Ilfracombe and Havant & Waterlooville.
Among the Kilmarnock trialists is Kyle Letheren, the former Swansea City, Barnsley and Plymouth Argyle goalkeeper who was signed on a short-term contract last season by Motherwell as cover for John Ruddy.
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