Cardiff chairman Mehmet Dalman expects the swift appointment of a "younger and offensive type of manager" to succeed Neil Warnock.
Warnock's Cardiff reign came to an end on Monday when he left his position by mutual position.
The 70-year-old was appointed in October 2016 with the club second from bottom in the Sky Bet Championship and guided them to promotion to the Premier League in 2018.
"Vincent (Tan) wants to take a much more hands-on involvement in the next manager, which is right," Dalman told BBC Radio Wales. "After all, he is the owner of the club.
"He wants somebody younger, an offensive type of manager.
"So the process is going on at the moment and we're down to a very small number of names now.
"I'd be very surprised if we don't announce something in the next 72 hours, certainly by the weekend."
Harris, who took Millwall into the Championship and had a short spell at Cardiff during his playing career, is the early favourite with the bookmakers.
Asked about Harris, who is out of work after leaving Millwall last month, Dalman replied: "Bookies know best.
"We're still going through the thought process.
"We will go through the names that we have and focus on one of them."
Cardiff are currently 14th in the Championship following relegation from the Premier League last season.