"As you start marking down the days before Thursday, it does mean something," he said. "There's a reason why they call it a deadline. We have to be mindful of what that means, where we are, what the math says about our chances."
The loss of Peavy will leave a void in the clubhouse, players said.
"His personality, the type of person he is, he came in here and fit in, not just on the field but off," first baseman Mike Napoli said. "We hung out. He has a great family, a genuine person who cares about everybody. A good friend."
The Red Sox confirmed that Allen Webster will start in Peavy's place on Sunday.
ESPN's Jim Bowden and The Associated Press contributed to this report.