The Merseyside club have struggled in all competitions this season as they sitting in the relegation zone after nine matches. Everton have collected eight points so far, one point behind Leicester, who are in the 14th spot.
Everton sacked Ronald Koeman on 24 October, a day after their 5-2 defeat to Arsenal at Goodison Park. The Foxes also parted ways with Craig Shakespeare just four months after making him their permanent manager.
Both the Toffees and the former Premier League winners are without a manager following their sacking of their respective bosses. Giggs, who is keen on taking the job at Everton or Leicester, is aware that he will face tough competition for these two positions.
"I think you look at those two teams, Leicester being champions two seasons ago, Everton being a fantastic club with a fantastic history, for me, these are clubs I would be interested in. However, there are a lot of coaches out there who would be interested in those positions," Giggs told Sky Sports.
Giggs was the assistant manager at United during Louis van Gaal's two-year spell at Old Trafford, which ended after the end of the 2015/16 season. Jose Mourinho replaced the Dutchman and the former Red Devils winger, who has been out of management since Mourinho's appointment at United, explained the conditions on selecting his next club.
"I've said all along that I'd be open to clubs which share the ambitions of myself. I want to improve clubs and improve players, to enjoy working for them and for players to enjoy the challenge," the Old Trafford favourite explained.
"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."