The left-back has signed with the German giants on a deal running until 2025.
The 23-year-old academy product has banked in excess of 100 appearances in all competitions for Reading since the start of the 2017 campaign and 41 of those have come in the Sky Bet Championship this season.
But the left-back has opted for a fresh start in Germany and has agreed a deal that runs until 2025 with Bayern, who have handed their incoming recruit the number three shirt previously worn by Xabi Alonso.
“My move to FC Bayern is a great honour for me, a dream come true,” said Richards, who made his only appearance for England Under-21s as a substitute in October 2019 in a 2-2 draw in Slovenia.
“I’m proud to be wearing the shirt of one of the world’s biggest clubs. I hope I can help the team continue to be successful in the future.
“My thanks to the management at FC Bayern for their trust. Our conversations were very convincing. I can’t wait to be on the pitch for Bayern.”
Reading say they offered Richards a new contract but he ultimately decided to head to the six-time European Cup champions when his existing deal expires next month.
Bayern’s board member for sport Hasan Salihamidzic said: “We’re delighted Omar is coming to FC Bayern.
“Omar is a technically gifted player on the left side of defence. He finds good solutions going forward, he’s very alert and we trust him to play a good role in our team.”