The 21-year-old, released by Saints at the end of the season, has the option of a further year should he impress.
Reeves told the club's official website: "It's been an interesting summer, seeing what's around.
"I was at Southampton for 12 years so it's been different looking around other clubs but I'm happy to be here in Milton Keynes.
"I feel I've definitely picked the right club. I love the way the team play and I'm really looking forward to being part of things here and hopefully help the team push on."