Mirallas suffered a recurrence of a hamstring injury in last weekend's win over Tottenham which had sidelined him for a month.
Hibbert has been struggling with a calf problem all season and Everton have now decided surgery is the best option to solve the problem, with the defender having an operation early next week.
"It looks like we'll lose Kevin for a few weeks," said Moyes.
"It's a really big blow - anyone who watched the game last week can see that.
"His hamstring isn't just right so it looks like we'll have to take more time over it so it will probably keep him out for a while.
"We thought we had managed it because it was quite a mild hamstring injury which we gave extra time because I was expecting him to be available for the midweek Arsenal game.
"I then didn't use him in the Manchester City game in the hope he would be right for the Tottenham game so we gave him extra days to make sure it was right.
"We now will have to take even more time unfortunately and it will be a big miss for us because he gives us a different attacking dimension from what we have and he'll be a loss.
"Tony Hibbert has a recurrence of his calf injury so he is going to have an operation on Monday which will keep him out for a while."
Source: PA
Source: PA