Baltimore claimed Willingham on Wednesday afternoon, according to the paper, meaning that the Orioles will have until Friday afternoon to agree to a trade with the Twins.
The Orioles (70-61), who are chasing their second consecutive postseason berth, have claimed several players during the trade waiver process, according to the Sun.
The struggling Twins (57-74) are more motivated to trade Justin Morneau, a free agent after this season, and have discussed dealing the first baseman to the Orioles, the newspaper reported.
But Willingham, who is under contract through next season, evidently is a more appealing target to the Orioles.
Willingham, 34, has struggled through an injury-plagued season in Minnesota, batting a career-low .212 with 12 home runs and 45 RBIs. The veteran outfielder enjoyed a strong season with the Twins in 2012, when he batted .260 with 35 homers and 110 RBIs -- both career highs.
The right-handed Willingham would be a welcome addition to the Orioles, who regularly play two left-handed hitters -- Nick Markakis and Nate McLouth -- in the corner outfield spots. Baltimore also has been lacking consistent production from its designated hitters throughout the season.