"We've seen this over the course of the last couple of years, that when this pops up, we need to be careful with him," Melvin said. "I don't know this is going to be a quick thing. This is a tough loss for us."
The A's also designated Cesar Valdez for assignment ahead of Wednesday's 7-4 loss to the Twins, promoting right-handers Bobby Wahl and Josh Smith from Triple-A Nashville to fill the two vacant bullpen spots.
Wahl made his Major League debut on Wednesday against the Twins, going one-third of an inning, allowing one run on two hits and also hitting a batter.
Doolittle's absence leaves Melvin with just one southpaw, Daniel Coulombe, in his bullpen, though he noted he has trust in right-handers Ryan Madson and Ryan Dull against left-handed batters, who were 0-for-12 with eight strikeouts against Doolittle.
Wahl, a 2013 fifth-round Draft pick by the A's who is ranked by MLBPipeline.com as the organization's No. 26 prospect, will be eased into action, according to Melvin. Smith, who was in camp with the A's as a non-roster invitee during the spring, will take on a long-relief role.
The 25-year-old Wahl had three saves and a 1.93 ERA in eight appearances with Nashville, and a 2.65 ERA and 14 saves in 45 games across three levels last year.
"One of the best feelings in the world," Wahl said of getting the call to the Majors. "It was a surreal moment. You dream of it your whole life, and when it actually happens, it's almost like, 'How did this happen?' It was really cool."
Wahl's wife, Kaitlin, who he's been with since high school, was his first call. She flew from their in-season home in Nashville to Minneapolis to be in attendance for her husband's potential debut.
"He's never got that call before," said Smith, who has not allowed a run over his last seven outings with Nashville. "To see the look on his face was really cool."
"We love what we saw from Bobby Wahl this spring, and we knew at some point in time he'd be here," Melvin said. "He was probably our top relief prospect here for the last year or so. It'll be great to see him out there, but [Doolittle] is a guy that pitched the eighth and ninth innings for us, so you can't really replace that. We'll just have to kind of maneuver things around a little bit."
Doolittle, who is in the fourth year of a five-year deal that includes club options for 2018-19, has allowed three runs in 7 2/3 innings for a 3.52 ERA this season.
"He works really hard," Melvin said. "He knows the issues he has, he maintains it, works diligently with it. Obviously we have our fingers crossed, but my guess is we're going to be conservative with him."
• Left-hander Sean Manaea (shoulder) is scheduled to throw an extended bullpen session at Target Field on Thursday. If all goes well, Manaea could be activated off the DL when eligible next week.