Valverde went 0-1 with a 5.59 ERA, converting nine saves in 12 chances. The Tigers will likely use Joaquin Benoit in most save situations for the time being.
Detroit has 10 days to trade, release or send Valverde outright to the minors.
"We have asked him if he would go to Triple-A for us, and try to work with him there, try to get him straightened out," general manager Dave Dombrowski said. "He did not give us an answer on that. It really doesn't make a difference at this point, since we would still have to get waivers at some time. He was open-minded to it. He didn't say no to it, but he needed some time."
The Tigers recalled right-hander Al Alburquerque from Triple-A Toledo to replace Valverde in the bullpen. They also put outfielder Matt Tuiasosopo on the disabled list with a strained left ribcage and recalled outfielder Avisail Garcia from Toledo.
The 35-year-old Valverde faltered during the 2012 postseason, and the Tigers were prepared to let him leave via free agency. But he didn't sign anywhere else, and Detroit eventually brought him back on a minor league deal.
He was called up to the majors in late April and allowed only one run through his first 12 appearances, but has not looked good at all lately.
"We think we have a guy that can still close games for us, if we get him fixed," Dombrowski said. "His velocity's been fine. Of course, he hasn't had his (split-fingered fastball) and he's had command problems. The last couple weeks haven't been good. Before that, he threw the ball fine, and that's what we're hoping that we can get back. We figure we have nothing to lose to try."
Detroit entered the season without much stability at closer when hard-throwing rookie Bruce Rondon failed to win the job during spring training. Benoit is 1-0 with a 1.80 ERA and four saves, and Drew Smyly has also pitched well out of the bullpen.
"Benoit, when available, healthy and ready to go and rested, will get most of the closing opportunities," manager Jim Leyland said. "And anything else, we'll mix and match."
Octavio Dotel has hardly pitched this year because of right elbow inflammation. Alburquerque was sent down because of control problems, but he has 26 strikeouts in 14 1-3 innings for Detroit, and the Tigers hope he can give the bullpen a lift now.
Dombrowski didn't want to look ahead to the possibility of a trade for help.
"We're comfortable with this group right now," he said.
The Tigers aren't the only team that could use some help in relief. In fact, Detroit won Thursday night's game on a ninth-inning homer by Jhonny Peralta off Boston's Andrew Bailey. The Red Sox are now giving Bailey a break from closing. Manager John Farrell said Koji Uehara would be the choice Friday night if a closer were needed.
"If you look out there right now, there are not a lot of shut-down, lock-down closers out there anywhere," Dombrowski said. "A lot of people have struggled with the back end of their bullpens."
Leyland said he hopes Valverde decides to stay in the organization, but he wasn't about to pressure him publicly.
"Whichever way he wants to go, I would respect," Leyland said.