Despite returning to first-team training last month, Schweinsteiger is surplus to requirements under Jose Mourinho at Old Trafford.
And despite interest from other clubs around Europe, 1990 World Cup winner Augenthaler has tipped his compatriot to swap England for the United States.
"I think he will move to the United States," Augenthaler told Omnisport. "I know he has offers from European clubs, in Turkey and in other countries.
"But I think he will go to the USA."
Schweinsteiger has watched United from afar this season, yet to make an appearance in any competition, having been forced to train alone for much of the campaign.
His move to Manchester on a three-year deal from boyhood club Bayern in 2015 had promised so much before injury and Mourinho's arrival as a replacement for Louis van Gaal derailed his career at United.
However, Augenthaler - a seven-time Bundesliga winner - insists Schweinsteiger's situation at United is part and parcel of football.
"Schweinsteiger wanted a contract before he went to United. He wanted one or two years more but Pep Guardiola planned for life without Basti, so he went to United," the 59-year-old added.
"United changed their coach. Van Gaal had to go and Mourinho doesn't like Schweinsteiger. That's football, it's business."