Schneiderlin signed on a four-year contract for a fee reported to be 25 million pounds ($38.9 million) while midfielder Schweinsteiger joined United on a three-year deal for a fee of around 14 million pounds.
Schweinsteiger, 30, came through the youth ranks at Bayern and spent 13 seasons with the Bavarians, winning the Bundesliga title eight times.
"I have thoroughly enjoyed my time at Bayern Munich, it has been an incredible journey and I didn't take the decision to leave lightly. Manchester United is the only club that I would have left Munich for," the midfielder told United's website.
"I feel ready for this new and exciting challenge in what I regard as the most competitive league in the world and I am looking forward to working with Louis van Gaal again."
Schweinsteiger made 536 appearances for Bayern since his debut in 2002, also winning seven German Cups and the Champions League in 2013.
The 25-year-old Schneiderlin was playing in the English third tier with Southampton five years ago but helped the club return to the Premier League and was in France's 2014 World Cup squad.
"I am delighted to be a Manchester United player. Once I learned that United were interested in signing me, it was a very easy decision to make," Schneiderlin said.
"The chance to be part of this squad to help this great club be successful was too good to miss."
United, who have also signed Italy right back Matteo Darmian and Dutch international forward Memphis Depay, travel to the United States for a pre-season on Monday.
($1 = 0.6437 pounds)
Image: Bastian Schweinsteiger
Photograph courtesy: Manchester United/Twitter
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