The left-handed slugger could provide another offensive boost for the Bucs, who are contending with the St. Louis Cardinals for the National League Central crown. They acquired hot-hitting Marlon Byrd from the New York Mets last week
The Pirates entered Saturday's game against the Cardinals in a tie with St. Louis atop the NL Central with about a month of the season remaining.
Morneau this year is batting .259 with a team-high 17 homers and 74 RBI for the Twins. Over 1,278 career games with Minnesota, the 2006 American League MVP has averaged .278 with 221 homers, 289 doubles and 860 RBI.
He leaves Minnesota third on its all-time homer list.
The British Columbia native was poised to have possibly his best season in 2010 when he suffered a concussion and never again regained his MVP form.
Presley comes to the Twins after hitting .264 with one double, one triple, two home runs and four RBI in 29 games for Pittsburgh this season.