Kilmarnock director Billy Bowie says the club has already received more than 20 applications for the vacant manager’s job.
Lee McCulloch, 39, left the position after a meeting with directors on Sunday.
Bowie hopes to begin interviews this week and has been pleasantly surprised by the level of interest, with some applications from outside the UK.
And he will keep an “open mind” but wants someone with “good experience”.
Kilmarnock, who lost 2-0 at home to Ross County on Saturday, sit bottom of the Premiership and face second-bottom Partick Thistle at Firhill after the international break on 14 October.
Speaking on BBC Radio Scotland’s Sportsound on Monday, Bowie said: “The main reason for that meeting was to look at what we can do to help improve the team’s performance.
“[Lee McCulloch] just felt like, ‘there’s not much more I can do here’. They’ve trained hard and worked hard and he just felt really let down with the performance on Saturday.
“The problem is that to be a manager, it’s hard to adapt from being in a personal level with people to actually being their manager.”
McCulloch was initially interim manager following Lee Clark’s departure in February and took on the job formally during the summer.
And Bowie, the club’s major shareholder, intends to have the new boss in place at Rugby Park in time for Killie’s trip to face Thistle.
“We’ve more or less received applications from all over the UK,” Bowie explained. “We will have exceeded 20 [applications] now and good calibre managers as well, some very experienced ones.
“Really, I feel we need a very strong-willed manager, very experienced and experienced in Premiership football; somebody that’s got a good bit of history behind him for carrying out the needs of Kilmarnock Football Club.”
Jim McIntyre left Ross County last week and Bowie said the 45-year-old would be “well worth an interview” should he apply for the position and the director indicated there were “two or three” outstanding candidates for the vacancy.
Bowie also indicated the club would be willing to pay compensation to appoint a new manager “as long as it’s reasonable”.
The new manager, when appointed, will be Kilmarnock’s ninth manager in eight years. Jim Jeffries left the club in January 2010, and was succeeded by Jimmy Calderwood, Mixu Paatelainen, Kenny Shiels, Allan Johnston, Gary Locke, Lee Clark and McCulloch respectively.
Bowie, the club’s major shareholder, has been on the board for the last three of those years.
“Allan was doing all the right things but just couldn’t get the results on the park,” explained the director.
“Then Gary came along and we started off best intentions and the results just weren’t coming forward and then Lee Clark came on board and Lee went away to Bury after that and that’s where Lee McCulloch came in.
“Anything that [the managers have] ever come to the board with, we’ve always helped support them. We’ve always found that it’s a fair request that they’ve asked for.”