Mee suffered a knee problem shortly before half-time in Tuesday’s 2-0 defeat against Leicester City at Turf Moor, and was replaced by Nathan Collins.
Boss Sean Dyche will check on the centre back on Friday, and explained: "We’re not too bad with other issues but Ben is going to be touch and go, we think, as to whether he makes it, and the risk and reward of that scenario.
"We’ll wait and see on more news tomorrow on him.
"It is more of a knock than a twist, which is favourable, but it’s whether he makes it in time, and whether it settles down in time.”
However, he will again be without Erik Pieters, Johann Berg Gudmundsson, Matej Vydra and Dale Stephens, although the latter two are closing in on a return to fitness and are likely to step up training next week.
Dyche said: "Erik won’t be involved, nor will Dale at the weekend.
"Erik is going to be a few weeks rather than days, and I don’t think Dale is serious, he could be training next week.
"Matej is just back on the grass and in and out with the group – we’ve not been having major training sessions because we’ve been recovering then they get ready for the next one, but hopefully he’ll get more training next week.
"We’ll wait and see with Johann, once he comes out of the boot and his calf gets moving again.”