Thompson, 25, pitched in two Grapefruit League games this spring, allowing one run over two innings. The non-roster invite was reassigned to Minor League camp on Monday.
Twins general manager Terry Ryan confirmed the suspension, and said he wasn't happy with Thompson violating the drug program. Ryan said Thompson tested positive for a recreational drug.
"We're going to keep him," Ryan said. "He's just suspended without pay. When you got 160 kids -- and this guy is not a kid -- there is no excuse. So we're going to take care of it. I'm disappointed in that."
Thompson made his Major League debut with the Pirates last season, posting a 7.04 ERA over 7 2/3 innings.