Graham, 53, was fired after the 2017 season, which ended a six-year stay with the Sun Devils. As an experienced head coach who has led multiple programs, he would fit what Kansas athletic director Jeff Long is looking for if he is focused on the rebuilding type for KU's next coach.
Graham is 95-61 in 12 seasons as a head coach, winning 10 games inside of his first two seasons both with Tulsa and Arizona State.
It is not clear if Graham is the top candidate at Kansas.
Long is now eight days into a search that started Nov. 4 when David Beaty was fired with three games left in his fourth season. Beaty, 6-40 at KU, is finishing the year with the team.
Graham has won at least 10 games in five of his 12 seasons as a head coach, leading his teams to bowls in 10 of those seasons.
He is known as one of the most progressive offensive minds in the country and a solid recruiter who brought N'Keal Harry -- named as a Biletnikoff Award (best receiver) semifinalist on Monday -- to Arizona State.
Graham's coaching tree includes seven current FBS head coaches -- Gus Malzahn (Auburn), Chad Morris (Arkansas), Major Applewhite (Houston), Jay Norvell (Nevada) and Billy Napier (Louisiana-Lafayette).
Beaty was Graham's receivers coach at Rice in 2006.
Graham was criticized for spending only one year at both Rice (2006) and Pittsburgh (2011). His stay at Arizona State was the longest of his career. The Sun Devils went 46-32 in his six seasons. Arizona State finished at least tied second place in the Pac-12 South in three of those six seasons.
There is no clear timeline on a replacement for Beaty.