Cleverley went under the knife to fix the long-standing issue and the 28-year-old hopes to return to action after the early weeks of the campaign.
"This is something I've put up with during my career, so it's good to get it sorted," Cleverley told Watford's website.
"I had been used to playing with pain in my ankle. Now that's going to be a thing of the past, I'll be fit and raring to win my place back once I've finished my rehab."
The former Manchester United youth academy graduate joined Watford last year, having failed to establish himself at Everton.
Cleverley has won 13 England caps, but hasn't played for his country since November 2013.