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Martial has missed five games with a thigh problem while Rashford sat out the Carabao Cup penalty shootout victory over Rochdale because of a groin injury.
Pogba was not able to take a penalty against Rochdale or attend a team meal on Wednesday after aggravating an ankle injury and could miss Arsenal’s visit to Old Trafford.
“I wouldn’t put my life on it [having Martial and Rashford available],” Solskjaer told a news conference on Friday.
“It’s part of the game, working with what you have.
“The situation [with Pogba] is he finished the game [against Rochdale] but he got a knock to his ankle which is very swollen so it was better for him not to go there [to the meal] with the team and he’s in a race for Monday
“Luke [Shaw] is back in training [on Friday]. He’s working really hard and we hope to see him before the international break. Maybe Monday.”
Injuries to Martial and Rashford could mean a first Premier League start of the season for 17-year-old Mason Greenwood, who scored for the second consecutive game against Rochdale.
The teenager is the only fit striker in the squad and Solskjaer insists he would have no problem picking him to play Arsenal.
“He’s proven when he’s been on the pitch — Astana and Rochdale — when you get him in the box he’s dangerous,” Solskjaer said.
“What’s pleased me about him is he doesn’t turn down chances to finish.
“He’s proved in those games he’s ready for it. I’ll have the weekend to think about it.”
United will line up against Arsenal — who have not won in the league at Old Trafford since 2006 — eighth in the table after winning just two of their opening six games.
Goals have been a problem with United only scoring more than once in a game on one occasion so far this season.
United have scored eight league goals — fewer than Norwich in 16th — and failed to score more than once in 17 of their last 20 games in all competition.
“You can see how much we’ve improved defensively, that;’s where we put the big money in,” Solskjaer said.
“We don’t concede many chances, concede too many goals.
“The other way going forward is where we’ve struggled with injuries. We’ve had some bad luck in that respect, about building relationships, doesn’t help when you lose players but sometimes the finishing could be better.
“I never said it was going to be easy this season. There are going to be ups and downs, highs and lows, and when we lose a game we have to trust ourselves and what we’re doing.”