The online bookmaker reported the stakes on the back of press speculation that the Germany international is the number one target of new manager Louis van Gaal.
Schweinsteiger previously worked with Van Gaal during the Holland manager’s tenure at Bayern Munich, with the pair enjoying a domestic double and a Champions League final run in 2009/10.
The 29-year-old is thought to be open to a move away from the Allianz Arena, although United are considered the only likely suitors in the betting.
Sky Bet’s Head of Football Trading Paul Wiggins said: “We’ve seen a flurry of bets on United signing Schweinsteiger this summer on the back of widespread speculation that he is Van Gaal’s primary target.
“It remains to be seen whether it is merely punters following the news or whether it is genuinely informed money, but we’ve been forced to cut his price significantly, having offered 5/2 on Friday morning.
“There seems to be little competition for his services at the moment, with Monaco next in the betting at 16/1 and attracting very little interest.”