Washington also purchased the contracts of right-handers Logan Kensing and J.D. Martin from Triple-A Syracuse. Martin is scheduled to make his major league debut in a start Monday night against the New York Mets.
The 36-year-old Tavarez was 3-7 with a 4.89 ERA in 42 games for Washington, his 10th major league team. In an 11-3 loss to the Chicago Cubs on Sunday, he allowed one run and got one out.
"It doesn't surprise me," Tavarez said. "They had to make some moves."
Tavarez is one of 17 relievers the Nationals have tried in their beleaguered bullpen, and he's made more appearances than any other pitcher.
"They made the right decision," Tavarez said. "I'll be with someone else -- soon."
Interim manager Jim Riggleman said Tavarez did the best he could.
"He's taken his lumps here," Riggleman said. "He feels bad about the lack of success he had."
The Nationals put Olsen on the DL retroactive to July 11. The left-hander, who is 2-4 with a 6.03 ERA, is scheduled to be examined by Dr. James Andrews in Birmingham, Ala., on Monday.
Kensing was 0-1 with an 18.00 ERA in six games with Washington earlier this season. In 23 games with the Chiefs, he was 1-1 with a 3.91 ERA and 12 saves.
Martin was 8-3 with a 2.66 ERA at Syracuse.