Queen of the South manager Allan Johnston is due back from a family holiday and his Killie namesake is keen to make a swift appointment.
Johnston told Press Association Sport: "I'm still hoping to have someone in place for Monday when the players return for testing. Our pre-season training schedule starts the week after and it's important we have someone in place before that."
Johnston confirmed the Doonhamers boss is among his leading candidates and says he hopes to discuss the job of replacing Shiels on Thursday.
"We have a number of strong candidates under consideration, all of them possessing different attributes," said the Rugby Park chief. "It is always good to see the level of interest that we have seen in the managerial position at Kilmarnock since it became vacant.
"We have received 40 applications. Some of them have come from older candidates who have worked at some very big clubs across Europe but are looking for a change of scene. Others are UK-based and younger. They are just starting out but have good coaching credentials. Allan would fall into that latter category."
The Queens boss enjoyed a successful first year in management as he led the Palmerston team to the Irn-Bru Second Division title and a Ramsdens Cup success. Rangers were beaten at Ibrox on the way to that cup triumph while former Hearts winger Johnston, 39, was named the PFA Scotland manager of the year.
And Killie chairman Johnston believes his club could be the ideal next stage for him to realise his coaching ambitions.
The Ayrshire club's chairman said: "Allan's only been in management a year but has proved himself by winning the Second Division championship and also beating Rangers with Queen of the South at Ibrox. He's a former Kilmarnock player, still lives locally and knows the Scottish scene. He knows how to work with a tight budget and young players and he's ambitious."
Meanwhile, the Kilmarnock chief also confirmed he has accepted an undisclosed fee from Cypriot champions Apoel Nicosia for striker Cillian Sheridan. The 24-year-old former Celtic attacker moved to Killie from CSKA Sofia last September but found himself on the fringes of Shiels plans after the return of Kris Boyd to Scotland.