Minaya was outrighted to Triple-A St. Paul on Wednesday, Dan Hayes of The Athletic reports.
Minaya was designated for assignment by the Twins on Monday after he posted a 5.59 ERA and 1.34 WHIP in 9.2 innings over six relief appearances in the majors to begin the year. After going unclaimed on waivers, he'll choose to remain with the Triple-A club rather than testing free agency.