The Twins will also receive a player to be named later or cash.
Morneau, 32, is batting .259 with 17 home runs and 74 RBI this season. He is a free agent at the end of the year.
The move should give the Pirates offense some power and allows them to move Garrett Jones to the outfield until Starling Marte returns from injury.
The busy Pirates also traded for outfielder Marlon Byrd and catcher John Buck of the New York Mets earlier in the week.
Minnesota gets Presley, who is batting .264 with two homers and four RBI in 72 at-bats this year.
Any player traded before midnight Saturday is eligible to play on a team's active postseason roster.
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The Cleveland Indians added outfielder Jason Kubel to their 25-man roster Saturday and optioned right-handed relief pitcher Carlos Carrasco to Class AAA Columbus.
The Indians acquired Kubel from the Arizona Diamondbacks on Friday.
The left-handed-hitting Kubel is batting just .220 with five home runs and 32 RBI over 89 games this season after a strong 2012 season.
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The Los Angeles Angels of Anaheim may soon drop the "of Anaheim" from their name.
The Angels are looking to negotiate a deal with the city of Anaheim that will keep them there through 2057. Part of the deal includes removing Anaheim from their official team name, the Los Angeles Times reported.
The club changed its name from "Anaheim Angels" to "Los Angeles Angels of Anaheim" before the 2005 season. The "Anaheim" name was required to appear in the team name to satisfy a lease provision. Under the new proposal, the team would officially be called the "Los Angeles Angels."
Anaheim's city council is expected to vote on Tuesday to negotiate the new deal with the club. The proposed agreement includes the team paying for a major renovation of Angels Stadium while owner Arte Moreno rents land surrounding the park for $1 per year and securing the development rights for 66 years.
Angel Stadium opened in 1966. It is the fourth oldest stadium in the major leagues behind Fenway Park in Boston, Wrigley Field in Chicago and Dodger Stadium in Los Angeles.
Angel Stadium was renovated in 1997. Then-owner Disney paid for much of the $118 million cost.
The proposal that keeps the team in Anaheim until 2057 does include opt-outs in 2036 and 2050.