Mbeumo, who remains Brentford's top scorer this season after netting his seventh goal at Brighton from the penalty spot on Wednesday, limped off the pitch after rolling his ankle and was replaced by Yoane Wissa before halftime.
"It is bad," Frank told the club website referring to the injury. "He'll have a scan later, but we don't know the exact number of weeks he will be out.
"He will definitely not play on Saturday, and he will be out for weeks. We will know more in the next couple of days."
Brentford will also be without Ivan Toney for another month as the striker cannot play for club or country until January 16th after being handed an eight-month ban after admitting to 232 breaches of gambling rules, which included betting on matches he played in.
"It was hard to replace Ivan, but we found a way. It's the same this time – we'll find a way," Frank added.
"Do I want Ivan and Bryan in my team? Any day: Yes. But that's football - it happens sometimes. We will find a solution and other players are ready to step up."
Cameroon international Mbeumo is one of eight first-team players currently sidelined with injury.
"Bryan's injury is a clear one – it's bad luck. He got caught, got pushed and rolled his ankle. It's very frustrating, but I'm a positive guy and we'll find a solution," Frank said.
"The positive news is that (Mikkel) Damsgaard got minutes (against Brighton) and is looking better and better.
"(Mathias) Jensen will be a decision I need to make in the next 24 hours, whether I want to push him to be ready or be a little more cautious and have him ready for (Aston) Villa I will assess him in the next 24 hours."
Brentford are 11th in the standings on 19 points from 15 games. They next travel to bottom side Sheffield United on Saturday.
