The 34-year-old has only once started three successive games for United since joining as a free agent in October. Having scored in the wins against Tottenham and Granada this week it was expected Cavani would start against the Clarets, but Marcus Rashford has returned in his place.
Solskjaer said the Uruguayan had told him he wasn't up to starting again.
But the United boss said Rashford has told him he's fit and ready to go as he continues to manage a foot injury.
"We've got to manage him well. He's not played or started three games in a week for a long time Edinson so when I spoke to him he felt being a substitute was the best thing," Solskjaer told MUTV.
"With Marcus, it's a long time since I've asked him how you are and he's said good. A very long time."