"Dennis is away from the team right now with his family," Donovan said before the Thunder played the Denver Nuggets.
Considered a leading candidate for the league's Sixth Man of the Year Award with averages of 18.9 points, 3.7 rebounds and 4.0 assists, the Thunder were anticipating his absence at some point and worked on how that would look during training camp.
They also got an unexpected preview of coping without their star backup when he was limited to just 22 minutes in the Thunder's 110-94 win over the Utah Jazz on Saturday, as Schroder was saddled with foul trouble.
"It happened this morning so he's with his family at this point and time," Donovan said. "I'll have a chance to probably talk to him and find out a little bit more, if it's not later today, tomorrow. But right now for at least this game, he'll be out. We'll probably be able to gather some more information once we get a chance to spend a little bit more time with him over the phone."
The 26-year-old Schroder and his wife, Ellen, welcomed their first child in February 2019.
"I'm not going to leave my wife by herself while she's having the second baby," he told reporters last month. "I'm for sure going to go there and support her and try as much as I can to be there for my family.
"I love my teammates. I love basketball. But family comes first all the time."
Donovan also said forward Terrance Ferguson will miss the Denver game because of a leg contusion.
Oklahoma City, the No. 6 seed in the Western Conference, entered Monday's game 1½ games behind Denver for the No. 3 spot.