Braehead Foods owner Craig Stevenson had hit out at a lack of communication from Kilmarnock chairman Michael Johnston over a bill of about £16,000 and threatened to send sheriff officers to Rugby Park with the petition.
But Stevenson announced he had come to an agreement with the club, who are close to £10million in debt, over the bill for their Park Hotel.
In a statement, Stevenson said: "I can confirm that Michael Johnston from Kilmarnock FC has now been in contact with debt recovery lawyers at MacRoberts in response to the winding-up demand.
"We have accepted an immediate payment schedule.
"We are happy Michael Johnston has communicated with us and that payment issues have been resolved."
No one at Kilmarnock was available for contact but Johnston had played down the issue on Saturday.
Johnston, who hosts the club's AGM on Thursday, claimed the bulk of the debt was not due until the end of the month and that there was no threat of a winding-up order.
He added: "The creditor is obviously looking to get some priority for payment for himself and it's unfortunate that he has decided to take it into the public domain."