Sports Illustrated first reported the news.
Miles, who hasn't coached since being fired by LSU in September 2016, would replace David Beaty, who said Nov. 4 that he was informed he would not return as coach after this season.
Kansas athletic director Jeff Long and Miles know each other from their time at Michigan in the late 1980s and early '90s. Long also tried to hire Miles when he was the AD at Arkansas.
Miles, who turned 65 last Saturday, is 141-55 in 15-plus seasons as a head coach at LSU and Oklahoma State.
Beaty, who agreed to finish out the season, is 6-31 with the Jayhawks, who are 3-8 this season after putting up a fight but ultimately falling to No. 6 Oklahoma 55-40 on Saturday night.
The Jayhawks haven't had a winning season or reached a bowl game since 2008, their penultimate season under Mark Mangino. After he was forced out, the school churned through Turner Gill and Charlie Weis before hiring Beaty, a nondescript assistant with ties to the program.
On Thursday, LSU announced it had reached a settlement with Miles for a one-time, lump-sum payment of $1.5 million on his remaining guarantee from the school. He still had been owed $6.5 million in guaranteed salary after being fired.
At the time of his dismissal, Miles was owed about $9 million, to be paid in monthly installments until 2023. But Thursday's agreement absolves LSU of any remaining funds owed to Miles, who coached the Tigers from 2005 to 2016 and went 114-34.
"It's time for both parties to move forward," LSU athletic director Joe Alleva said in a statement. "One of the challenges of the buyout that was in place was there just wasn't a lot of incentive to move on to other things. We were looking to provide that, and Coach Miles and his representatives also were ready."
ESPN's Chris Low, Adam Rittenberg and Dan Murphy and The Associated Press contributed to this report.