Morneau, 32, signed a two-year, $12.5 million contract with the club during the offseason to replace the retired Todd Helton at first base.
Last season, Morneau hit .259 with 17 home runs and 77 RBIs between the Twins and Pirates. He went 7-for-24 (.292) in six postseason games.
Second baseman Josh Rutledge did not play at all Monday because of a recurrence of left ankle soreness, and his availability is also day to day.
Thomas Harding is a reporter for MLB.com. Follow him on Twitter @harding_at_mlb. This story was not subject to the approval of Major League Baseball or its clubs.