The right-back was forced off after sustaining the injury in the second half of last Saturday’s 0-0 draw at Brighton.
Ahmed Elmohamady is now set to make just his second Premier League start of the season when Leicester City visit Villa Park on Sunday.
Boss Dean Smith said: “Matt has a hamstring strain. I am not sure the grade but it will mean a few weeks out, which is a blow for us.
“But it is something we have been good at dealing with so far. We have not had too many injuries but when we have people have come in and done the job for us.”
Cash has enjoyed an excellent first season in the Premier League since joining from Nottingham Forest for an initial £14million last summer.
“He is one of our new signings who has hit the ground running,” added Smith. “If there were fans in the stadium at the moment he would be well on the way to being a favourite.”