Garcia's grandfather passed away in Venezuela, and the Yankees said that Garcia will be away from the team for several days.
Igarashi, 33, was claimed on waivers from the Blue Jays on May 29. He allowed four runs in two appearances for Toronto this year and made 79 appearances over the last two seasons for the Mets.
"He's a guy that can give you up to 50 pitches, but he can also be a one-inning guy," Yankees manager Joe Girardi said. "He's kind of a power guy. He's going to get an opportunity over the next few days."
Igarashi was 1-2 with a 1.73 ERA in 22 appearances at the Triple-A level this year, three of which came in the Yankees' system for Scranton/Wilkes-Barre. In 21 innings, he struck out 39 and walked six.