Rodgers, the son of Liverpool manager Brendan, was released by the Latics at the end of the last campaign, but the former Chelsea youth player is delighted to be moving to a club closer to home.
The 21-year-old told the club's official website: "I'm from down the road in Reading so it was something that I was interested in doing and I'm glad it's done now.
"I started my professional career at Brighton before moving to Oldham. I was close to my family there, with my dad at Liverpool but I just wanted to move closer to home.
"Thankfully, Swindon were interested and it was something that I could do."
Rodgers made just 14 appearances during his time at Boundary Park.