Mee was a regular fixture in the Brentford side last season, starting 37 of the club's 38 Premier League matches.
The defender, who helped the Bees achieve a top-half finish, won the supporters' Player of the Year award for his contribution last term.
Mee has been unable to have the same impact this term after seeing the opening stages of the season hampered by injury.
After missing the opening three top-flight matches with a calf injury, Mee returned to the side for league fixtures against Bournemouth and Newcastle United.
However, the 34-year-old has struggled with a muscular issue in recent weeks, forcing him to miss league games against Everton, Nottingham Forest and Manchester United.
Following a spell on the sidelines, there is hope that Mee could return to action for Saturday's clash with his former side Burnley.
"Mee is getting closer so that is a positive", Brentford boss Thomas Frank told reporters in his pre-match press conference.
"He could be available for selection. It's still only four weeks after his injury, but he's been training with the team."
Frank has also received a boost with the news that goalkeeper Mark Flekken has recovered from appendicitis and is in line to feature against the Clarets.
As well as providing positive news on Flekken and Mee, the Brentford head coach gave an update on a number of injury absentees.
Frank added: "Shandon Baptiste and Myles Peart-Harris have both part-trained with the team, and Josh Dasilva is on his own but on the grass. Mikkel Damsgaard his progressing really well after his minor knee surgery.
"Keane Lewis-Potter is progressing quite well. I don't know the timeline exactly, but it's not as long as I had feared. It's positive."
Brentford are also without Kevin Schade and Rico Henry, while Aaron Hickey is set to serve a one-match suspension for an accumulation of yellow cards.
The Bees, who occupy 15th position, are hoping to end their six-game winless league run and claim their second top-flight victory of the season.