Johnston could leave Queen of the South after just one year as manager.
Johnston, 39, and former Blackburn boss Henning Berg have been interviewed for the managerial vacancy at Kilmarnock, who are also keen on Sheffield United coach David Unsworth.
Johnston, who spent five years with Kilmarnock as a player after two and a half seasons with Sunderland in the late 1990s, had a brilliant first campaign as manager, steering Queens to the Division Two title and the Ramsdens Challenge Cup, including an early-round win at Rangers.
He was voted PFA Scotland’s manager of the year.