Tate hasn't played since the fourth quarter of a loss to the Saskatchewan Roughriders on July 5.
Calgary's No. 1 quarterback suffered a setback after throwing the ball in practice earlier this week.
Hufnagel compared Tate's injury to tennis elbow and said his quarterback was experiencing pain in soft tissue near the elbow of his throwing arm.
Kevin Glenn will be Calgary's starting quarterback in Friday's rematch with the Roughriders.