The Dutchman went off injured during Edinburgh's RaboDirect Pro12 win over Treviso on Friday night.
A subsequent MRI scan conducted at the Murrayfield Hospital on Monday showed further damage to his ankle ligaments, which will require consultant surgeon John McKinley to operate.
Edinburgh Rugby head coach Alan Solomons said: "This is really disappointing news for Tim. We're very sorry he'll miss the viagogo Autumn Tests.
"He's an asset for Edinburgh Rugby and Scotland. We all wish him a speedy recovery and look forward to seeing him back scoring tries again soon."
Visser wrote on his official Twitter account: "So the bad news is that I broke my Fibula and damaged some ligaments. Operation soon and hopefully be back in three to four months."