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Bailly has missed all four of United's games since the March international break after testing positive for Covid-19 while away on duty with the Ivory Coast and stayed in the West African country while recovering.
The 27-year-old is back in Manchester but speculation over his long-term future has suggested that he is unhappy with a lack of playing time under Solskjaer, having largely provided cover to first-choice centre-halves Harry Maguire and Victor Lindelof this season.
Bailly's current contract at Old Trafford is set to expire at the end of next season but Solskjaer said that talks over a new deal are ongoing and that he hopes to have him back fit and available on Sunday.
“Eric has been isolating back home in the Ivory Coast. He's back to England now. We hope he can be available for the weekend," Solskjaer said.
“Hopefully he can train tomorrow, depending on how he feels within himself. He's had his tests today so we need to get the confirmation that he's still negative.
On Bailly's future, the United manager added: "Eric has played well when he's played. I know there's been speculation. We're talking to him about his contract so I don't see any issues really."
Solskjaer also hopes that Marcus Rashford will be able to play a part against Burnley having sat out of Thursday night's comfortable 2-0 win over Granada in the Europa League quarter finals.
Rashford has been carrying an ankle injury since last month's Manchester derby and has had his minutes managed by Solskjaer over the past few weeks. The 23-year-old was an unused substitute on Thursday, though was named on the bench in case he was needed.
“A big bonus that he didn't have to put any pressure on his foot," Solskjaer said. "I've got to hope that he can be available and not make it worse. That's the decision I've got to make, fingers crossed.
“He wasn't due to be on the bench today really. He didn't train yesterday, but just in case he put his hand up and said if needed he could do a job. It's a decision I've got to make on Sunday morning but he's important for us.
“It's a difficult one because you've got to go by how Marcus feels in a couple of days.”