The Uruguay international sustained the injury during Saturday's 2-1 defeat by Atletico Madrid after a heavy challenge from winger Lee Kang-in, an incident that did not result in a booking.
Valverde completed the match but was later found to have suffered significant trauma to the ankle.
Real manager Jose Mourinho criticised the officiating after the game, questioning why no penalty was awarded for the challenge.
A grade-three sprain is the most serious type of ligament injury and involves a complete tear of the affected ligament.
Spanish media reports said Valverde is expected to be sidelined for between two and three weeks, ruling him out of Uruguay's upcoming international fixtures.