Williams, who has been capped 12 times by the USA at the age of 24, will sign a four-year contract on July 1, subject to international clearance.
He becomes the Royals' third signing of the summer - following former Real Madrid winger Royston Drenthe and England international Wayne Bridge - following their relegation from the Premier League.
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"I spoke to the manager yesterday when I did my medicine check and he is a really great, nice guy. I'm looking forward to working with him, everything is perfect here.
"I spoke to Gylfi [Sigurdsson] also because he is a good friend from Hoffenheim and a hero here!
"I would like to say to the Reading fans I will give 100% every game, I am proud to wear this jersey and will do everything to help this team back to the Premier League."
Director of Football Nick Hammond added: "Danny is a versatile player, he possesses great energy and already has great experience in one of the top leagues in Europe.
"He is an international player and at 24 years of age still has plenty of development ahead of him, so we will be delighted to welcome him to the club."
Chairman Sir John Madejski continued: "I remember American players Marcus Hahnemann and Bobby Convey doing very well in a Reading shirt in recent years and I am certain Danny will do the same.
"We have seen many players grow and flourish at Madejski Stadium and we wish Danny every success in following that path."