The 26-year-old will officially link up with the Royals once his contract at Oakwell expires on July 1.
Yiadom had been with Barnsley - who will play in League One next season - since the summer of 2016, having previously turned out for Hayes & Yeading, Braintree and Barnet.
The right-back's form at club level saw him earn a first Ghana call-up at the end of 2016, before going on to make his international debut in the Africa Cup of Nations in early 2017.
"I'm very happy that Andy is joining Reading as my first signing," Reading manager Paul Clement told the official club website.
"He is someone that I've known for a while, and whose progress I was tracking when I was at Swansea.
"Andy comes here with good experience of the Championship - he's a good age at 26, and he has the right desires and ambitions which will help us to move forward with this club."
Yiadom had been linked with a number of Premier League clubs, including Huddersfield Town, in the January transfer window.