But the skipper, and left back Erik Pieters, could be back in contention for Easter Sunday’s game at West Ham.
Mee has missed the last four games with a hairline fracture to his fibula, suffered in the first half of the 2-0 defeat at home to Leicester at the beginning of March.
And Pieters has been out with a knee problem suffered in the draw at Crystal Palace in February.
However, there is encouraging news on the pair, with nine games to play as Burnley fight for their Premier League status.
Boss Sean Dyche said: “Ben and Erik are beginning to make progress.
“Not (available) for this week, but for West Ham, we will wait and see.
“They are with the physios and sports science, and then it is how quickly they can get out with us, so there is a possibility.”
Check out the latest Premier League transfer speculation below...
![West Ham will have to pay up to £66.7m to sign Lucas Ocampos from Sevilla. (Estadio Deportivo)](https://www.thestar.co.uk/jpim-static/image/2022/04/08/21/GettyImages-1389568703.jpg?crop=3:2&width=800)
1. Ocampos valuation revealed
West Ham will have to pay up to £66.7m to sign Lucas Ocampos from Sevilla. (Estadio Deportivo)
![Aston Villa have joined Leeds and Tottenham in the race to bring Noni Madueke to the Premier League. (talkSPORT)](https://www.thestar.co.uk/jpim-static/image/2022/04/08/21/GettyImages-1390132487.jpg?crop=3:2&width=800)
2. Madueke in demand
Aston Villa have joined Leeds and Tottenham in the race to bring Noni Madueke to the Premier League. (talkSPORT)
![Manchester United have made an approach for Everton ace Richarlison but face competition from Real Madrid. (Sport Witness)](https://www.thestar.co.uk/jpim-static/image/2022/04/08/21/GettyImages-1390033515.jpg?crop=3:2&width=800)
3. Man United target Richarlison
Manchester United have made an approach for Everton ace Richarlison but face competition from Real Madrid. (Sport Witness)
![Middlesbrough are battling Burnley and Norwich City for the signing of Chesterfield striker Kabongo Tshimanga. (Football League World)](https://www.thestar.co.uk/jpim-static/image/2022/04/08/20/GettyImages-1361576754.jpg?crop=3:2&width=800)
4. Battle on for Tshimanga
Middlesbrough are battling Burnley and Norwich City for the signing of Chesterfield striker Kabongo Tshimanga. (Football League World)