Mount and Varane have rejoined training following spells on the sidelines with injuries but weren't fit enough to make the trip to Germany ahead of the 4-3 defeat at Bayern Munich in the Champions League on Wednesday.
There is hope, however, that the pair can be named in the squad for the game at Turf Moor this weekend if they come through a final training session on Friday unscathed.
The United squad stayed in Munich overnight and trained at Bayern's academy on Thursday before flying back to Manchester. Ten Hag was without 12 senior players against Bayern and was forced to name three goalkeepers on the bench at the Allianz Arena.
Having Mount and Varane back against Burnley will be a boost to the Dutchman but there are fears two other injured players, Luke Shaw and Aaron Wan-Bissaka, could be facing lengthy spells out of action.
Shaw hasn't played since the 2-0 defeat to Tottenham Hotspur in August because of a muscle problem and, according to a source, there are concerns he will not be back before the November international break, which would mean at least another two months out.
Wan-Bissaka could also be facing a similar spell out after picking up a hamstring injury against Brighton. United have confirmed the right-back will miss "several weeks" but it could be more than two months.
United face Burnley on a run of three straight defeats in all competitions and four in six games so far this season. Ten Hag's team have conceded 12 goals in their last four games including at least three in their last three matches.