Busch was sanctioned earlier this month after making threatening comments to a reporter following a Nationwide Series race at Dover. That violated a probation from a previous disciplinary incident at Darlington.
Finch said Busch would continue to drive Phoenix's No. 51 Chevrolet after his suspension ends Wednesday and would get back behind the wheel in this weekend's 400-mile race at Michigan International Speedway.
"At the end of the day, we are racers so we're going racing together with Kurt and the No. 51 Phoenix Racing Chevrolet," Finch said. "We know adjustments have to be made, but how we fix that is between Kurt and myself.
"We're going to go to the track, work hard, race hard and work on trying to attract a sponsor, and we're going to do that together."
David Reutimann, who drove the No. 51 car Sunday at Pocono Raceway, will return to the No. 10 car for Tommy Baldwin Racing at Michigan.