Because Villarreal refused the assignment, the Astros will either have to trade him or release him, assuming no team claims him.
"We don't know much more than that at this point," Houston general manager Ed Wade said. "We'll probably run out the entire 10 days before it's resolved."
The Astros are responsible for paying the balance of Villarreal's contract, which totaled $2.85 million when they signed him to a two-year deal last winter.
The Astros were hoping Villarreal would accept the assignment to Triple-A Round Rock, where he could receive more regular work and try to get back on track after posting a 5.02 ERA over 35 relief appearances.
When the 10-day window runs out, Villarreal will be a free agent and can sign with any of the other 29 Major League teams. In the interim, he's on his own.
"I was told he's in Monterrey, Mexico, working out with the team he used to pitch for," Wade said.
In other roster news, the Astros still are awaiting word on if Reggie Abercrombie has accepted his Minor League assignment. The outfielder was designated for assignment on Sunday, after the Astros lost to the Braves in 17 innings. Abercrombie was removed from the roster to make room for right-hander Chad Paronto.
Alyson Footer is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.