Manchester United great Giggs served as an assistant to Louis van Gaal during the Dutchman's Old Trafford tenure, leaving when Jose Mourinho took the reins in 2016.
The former Wales international also interviewed for the job at Swansea City last season but has only taken charge of four senior matches when placed in caretaker charge of United after David Moyes' departure in 2014.
Everton sacked Ronald Koeman on Monday after a 5-2 weekend defeat to Arsenal at Goodison Park left them in the bottom three — a point worse off than Leicester after caretaker boss Michael Appleton oversaw a 2-1 win at Swansea in the wake of Craig Shakespeare's dismissal.
"For me, these are clubs I would be interested in," Giggs told Sky Sports.
"I think you look at those two teams, Leicester being champions two seasons ago, Everton being a fantastic club with a fantastic history.
"However, there are a lot of coaches out there who would be interested in those positions."
Giggs is keen to be the latest student of Alex Ferguson to go into management but, despite his lofty current targets, he would not necessarily insist upon a Premier League job.
"I've said all along that I'd be open to clubs which share the ambitions of myself," he added.
"I want to improve clubs and improve players, to enjoy working for them and for players to enjoy the challenge.
"There are clubs in the Premier League which would be good jobs but also in the Championship and League One.
"It's more about the philosophy for me than the stature of the club."