Terms were not disclosed, but the Boston Celtics own only the biannual exception ($1.975 million) to match.
The Timberwolves' offer was expected to trump that, meaning Stiemsma is almost certainly bound for Minnesota.
Stiemsma received consistent interest from the Timberwolves this summer, meeting with them near the start of free agency and waiting patiently as they sorted out their own cap issues while pursuing front-line players.
The Celtics would have three days to match an offer sheet, or they could simply rescind their $1.1 million qualifying offer to allow Stiemsma to sign as an unrestricted free agent.
Stiemsma, a Wisconsin native, heads closer to home with a chance to play again for the team that gave him his first NBA opportunity.
Stiemsma signed with the Timberwolves at the end of the 2009-10 season, but was waived before appearing in a game.
After stints in South Korea, Turkey and the NBA Development League, he landed in Boston after the lockout and earned a roster spot. By the end of an injury-decimated 2011-12 season, he was Boston's first big off the bench.
Stiemsma averaged 2.9 points, 3.2 rebounds and 1.5 blocks over 55 games with the Celtics.