Alford, a second-draft pick in 2013, was injured last month at Carolina. Coach Mike Smith said Alford underwent surgery on Monday.
Smith added that the Falcons (5-9) hope to have Julio Jones, the NFL's second-leading receiver, "ready to go" Sunday at New Orleans.
Jones suffered a hip injury late in a loss at Green Bay two weeks ago and didn't play in Sunday's home loss to Pittsburgh.
Strong safety William Moore didn't dress against the Steelers because of a foot injury he suffered against the Packers.
Right guard Jon Asamoah left the Pittsburgh game in the second half with a back injury.
Smith wasn't sure if the injured players will practice this week.
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