Speaking on WFAN, Boone said Montgomery, who was placed on the disabled list, won't throw for two to three weeks.
"It's not great news, but not worst-case news, either," Boone said. "And hopefully he can use this time to get well, get strong and help us for most of the second half."
Montgomery exited Tuesday night's start against the Houston Astros after throwing just seven pitches.
Montgomery flew back Wednesday to New York, where he was evaluated by Dr. Chris Ahmad.
Asked right after the game about how high his level of concern over the injury was, Montgomery said "not very high." The Yankees also had been hoping the injury wasn't significant.
Though a flexor strain can be a precursor to a Tommy John injury, there's "nothing that would suggest" that at this time for Montgomery, Boone said.
Montgomery could be seen during his lone inning of action flexing his left arm as if experiencing some discomfort. Asked to describe exactly where he was injured, he pointed to an area of muscle just below his elbow on the back of his arm.
The problem had been nagging Montgomery over the past two days, but the lefty thought he could "kind of throw through it" as he underwent additional exercises and extra treatment. He said he has had forearm tightness in the past, but he wasn't sure if this was the exact same location as those injuries.
Montgomery (2-0, 3.26 ERA) drew praise from his manager Wednesday.
"He's pitched really well," Boone said. "He's kind of given us a chance to win every time he's gone out there. So we'll just have to pick up the slack. In the meantime, Domingo German will go into his spot and make his start on Sunday."
To replace Montgomery on the 25-man roster, the Yankees signed minor league pitcher David Hale to a major league contract, selecting him from Triple-A Scranton/Wilkes-Barre.
In addition to that move, the Yankees also transferred outfielder Jacoby Ellsbury (plantar fasciitis) to the 60-day DL.
Outfielder Clint Frazier, who had been rehabbing in the minors, was officially reinstated from the seven-day concussion DL and optioned to Triple-A. Frazier, who missed most of spring training with the head injury, had spent the past two days on rehab assignments in Triple-A. He hit a home run Wednesday for the RailRiders.