The Championship club are in 'advanced talks' with McDermott, who was sacked by Reading last month, and bookmakers have stopped taking bets.
"The process of appointing a permanent replacement for Neil Warnock has progressed significantly this week and when all matters are finalised we will be in a position to make a formal announcement," said Leeds in a statement.
"Neil Redfearn will remain as caretaker manager until such time when an appointment is made, and he is being kept informed of the situation by the club's board."
McDermott, 52, spent more than three years as boss of Reading, which was his first major job in management having joined the club in 2000.
He steered the Royals to the Championship title last season - only the second time the club has reached the top flight - but found himself dismissed with the Royals drifting towards relegation.
The Sky Sports presenter David Jones tweeted: "Brian McDermott has pulled out of tonight's Footballers' Football Show. I'm told he has a very good reason."
McDermott's first task is keeping Leeds in the second tier after an alarming slide towards the bottom of the table.
Warnock left Elland Road 10 days ago following the home defeat to Derby, while a run of four straight losses has seen Leeds slip to within five points of the bottom three with five games remaining. Neil Redfearn had been placed in temporary charge.
Should the club stay up, McDermott will be required to prepare Leeds for a successful promotion push next season after a disappointing 2012/13 campaign.
In a strange twist, Leeds Rhinos rugby league team are already managed by a Brian McDermott.