Manager Ned Yost said he likely would go with Chris Young in another spot start this Sunday to take Vargas' place.
Vargas agreed with the decision to put him on the DL. He last pitched Tuesday against Cleveland, and he gave up just two hits and two runs in six innings.
"After my bullpen [session] the first time I pitched against Cleveland [April 27], I noticed some forearm tightness that wasn't there before," Vargas said. "We got it under control the first few days, and then we switched in the rotation with [Jeremy] Guthrie to buy a couple extra days, but then it definitely got fatigued [Tuesday].
"Nobody's going to feel great all the time, but it wasn't a normal thing to deal with. We took a look at it the next couple of days, and then decided to get ahead of it and get it under control and not to have to deal with it the rest of the year."
Vargas said he plans to test the flexor muscle, which protects the ulnar collateral ligament, next week in a throwing session, and he believes he will be on the disabled list for the minimum amount of time.
"We're anticipating just the two weeks," Vargas said. "I'm anticipating taking the 12 or 13 days I got left and get the inflammation out of there and get the flexor firing again properly."
"At the end of the year, we probably don't put him on the disabled list," Yost said. "But this early in May, it just makes sense to nip it in the bud."
Jeffrey Flanagan is a reporter for MLB.com. Follow him on Twitter at @FlannyMLB. This story was not subject to the approval of Major League Baseball or its clubs.