The Cameroon international has not played since December after sustaining an ankle injury during a 2-1 defeat to Brighton & Hove Albion.
Mbeumo was crucial to Brentford's efforts in the first half of the season, scoring seven goals and providing four assists in 15 Premier League appearances.
In his absence, the Bees have won just twice in 13 Premier League matches, leaving them in 15th position and five points above the drop zone.
Mbeumo, who scored in the reverse fixture against Burnley, is in contention to feature in Saturday's matchday squad, although Frank is keen to exercise caution with the 24-year-old.
"He's very close," the Brentford boss told reporters. "I have a decision to make, to maybe involve him in the squad this weekend.
"He's been on the grass for a week, he looks good, and he's one of our best players. I need to find out if it's just a touch too early, or to bring him for the game against Burnley."
Tottenham Hotspur loanee Sergio Reguilon is expected to return to the fold after missing the recent defeat to Arsenal with a hamstring issue.
However, Frank has confirmed that captain Christian Norgaard will be absent due to a back injury he sustained at the Emirates Stadium.
Frank said: "[Norgaard] is not available for the squad and that's also why he's not going away with Denmark. "We'll take it week by week, hopefully it's not a long one."
The Bees are also still having to cope without the services of Josh Dasilva, Ben Mee, Rico Henry, Ethan Pinnock, Kevin Schade and Aaron Hickey.
Having seen their campaign hampered by a lengthy injury list, Frank has admitted the club are looking into the reasons behind their continuous fitness issues.
"We have definitely had too many injuries," Frank added. "We are looking into it, we have to look into it. We are having a very open discussion with the medical department. They've been working very hard to get players back.
"We all want to minimise injuries. We all want the same. We've had eight contact injuries, and Christian is another. We've been extremely unlucky."
After taking just one point from their last five matches, the Bees will travel to Turf Moor with hopes of claiming their first win in over a month.