Zaire's planned visits were first reported by IrishIllustrated.com and confirmed by ESPN.
Zaire, who is expected to graduate this month and said last week he will play his fifth season elsewhere, is expected to visit the Badgers on Wednesday and the Tar Heels on Friday.
UNC redshirt junior quarterback Mitch Trubisky has another year of eligibility, but he could declare early for the draft, as he is Todd McShay's No. 1 quarterback prospect.
Pittsburgh is also in the running for Zaire's services, the source said, while adding that Auburn expressed some initial interest but would be a long shot given Zaire's desire to avoid a zone-read offense in favor of a pro-style set.
Zaire, who a source said last week was granted an unconditional release from Notre Dame, will be immediately eligible wherever he plays next season. That would include UNC, despite the Heels being on the Irish's schedule in 2017.
Zaire won 2014 Music City Bowl MVP honors in his first career start and started the first two games of Notre Dame's 2015 season before suffering a broken right ankle that ended his season. He worked his way back from injury in time for 2016, and coach Brian Kelly said in preseason camp that Zaire and DeShone Kizer would both play this fall.
Kizer assumed starting duties after both played in the season-opening loss to Texas, however.
Zaire will leave the Fighting Irish with a career 59.2 completion percentage for 816 yards, six touchdowns and zero interceptions. The lefty added 324 rushing yards and two rushing touchdowns. He finished with a 3-0 career record as a starter.