Zinchenko, 29, suffered the injury in the third minute of Saturday's 4-1 Eredivisie win over Fortuna Sittard. The Ukrainian defender was fouled by Fortuna winger Dimitris Limnios, whose tackle made Zinchenko's leg buckle.
"Examination at the hospital has shown that surgery is required. This will take place in the near future. The 29-year-old Ukraine international will then face a lengthy rehabilitation period, which means he will not play again this season," the club said in a statement on Monday.
The experienced left-back arrived from Arsenal in the last week of January and signed a short-term deal in Amsterdam, where his contract will run out in the coming summer. The six-month contract allowed new technical director Jordi Cruyff to examine whether or not Zinchenko would be given a longer contract.
Zinchenko debuted for Ajax on February 8th against AZ, playing 15 minutes in Ajax's 1-1 draw in Alkmaar. Against Fortuna Sittard, Zinchenko was taken off after just six minutes.
Ajax are currently in third place in the Eredivisie, 17 points behind league leaders PSV.