The 21-year-old suffered the injury during last week's 1-1 draw with Brentford at the Macron Stadium.
It was initially thought the the youngster would be ruled out for the remainder of the campaign, but the Trotters retain hope that they can get the former Crewe Alexandra man back to full fitness for the closing weeks of the season.
After visiting a specialist in London, Clayton has now gone under the knife and will continue to rest before beginning his rehabilitation.
Speaking to the club's official website, boss Neil Lennon said: "It's a blow for Max and of course for the squad. He's a great lad and we'll be with him every step of the way throughout his recovery period."