Damsgaard came off the bench to feature in Brentford's opening two Premier League matches against Tottenham Hotspur and Fulham.
However, the Denmark international has not featured since the 3-0 draw with Fulham on August 19 due to an injury issue.
As a result, the 23-year-old has missed Brentford's last six matches across all competitions, including the midweek EFL Cup defeat to Arsenal.
Speaking ahead of Sunday's league meeting with Nottingham Forest, Frank revealed Damsgaard has undergone a minor knee operation.
"He saw a knee specialist and had a minor operation this week," the Brentford boss told reporters at Friday's press-conference.
"Everything looked good in terms of the bigger parts that we could be worried about and that's positive. They cleaned the knee and hopefully that's the bit that will help him.
"We hope he will be back soon, but there is no timeframe. We will see how things settle down and then we will see how close he is."
Damsgaard's teammates Rico Henry and Kevin Schade also underwent surgery on their respective injuries during the week.
Henry is set to miss the remainder of the season after sustaining an anterior cruciate ligament injury earlier this month.
Schade, meanwhile, is expected to be sidelined for several months after having an operation on an adductor injury.
Ben Mee, Shandon Baptiste and Josh Da Silva are also in the treatment room, leaving Frank with a lengthy injury list ahead of Sunday's trip to the City Ground.
After failing to win their last four Premier League matches, the Bees will look to return to winning ways on Sunday.