The pair were able to train at Carrington on Monday but will be assessed on Tuesday after only joining in on part of the session.
Solskjaer said he is hopeful Victor Lindelof and Luke Shaw, who missed the 1-0 win over Watford on Saturday because of injury, will be available to feature at Turf Moor after training as normal on Monday.
"I've a decent amount of players," Solskjaer told a news conference on Monday.
"We'll see with Eric and Paul how they are tomorrow [Tuesday]."
Victory over Burnley will see United go clear at the top for the Premier League table after 17 games for the first time since 2012 but Solskjaer has urged his players not to get carried away.
"We are better off than at the same time last year," he said.
"We've won more, scored more and played better football. That's a positive, but nobody remembers January tables."
The fixture at Turf Moor is United's game in hand after starting their campaign a week later because of their Europa League commitments last season.
The delayed start to their season means they have been forced to play catch up and Solskjaer admitted it has been tough "mentally" on his players.
"It made it more difficult for players," he added. "A game in hand is only good if you win it. You need to put the points on the board. We had a difficult start against Crystal Palace and Tottenham and it made it harder for the players mentally to look at the table."
Solskjaer is expecting "two or three" squad players to leave the club in January but said he does not think there will be any additions before the deadline at the end of the month.
"January is always difficult but if something comes up that you think it's a long-term thing, that's another scenario," he said.
"It's unlikely something will happen. Two or three might go out as they deserve to play more football for the good of their own careers."