A source told ESPN's Adam Schefter that Alford is expected to be sidelined for six to eight weeks.
After Arizona's 33-26 preseason loss to the Oakland Raiders on Thursday night, coach Kliff Kingsbury described the injury as being to Alford's lower leg and said he would miss an "extended amount of time."
NFL Network first reported the nature of Alford's injury.
Alford's absence leaves the Cardinals extremely short-handed at cornerback to start the season. Alford was expected to start opposite eight-time Pro Bowl cornerback Patrick Peterson, but Peterson was suspended the first six games of the regular season for violating the NFL's policy on performance-enhancing drugs.
Alford could be back right around the time Peterson is eligible to return for the Cardinals' Week 7 game against the New York Giants. In the meantime, it's likely that rookie Byron Murphy and Tramaine Brock will be the Cardinals' starting cornerbacks in Week 1 against the visiting Detroit Lions.
The 30-year-old Alford signed a three-year contract worth $22.5 million with Arizona in February.