It was a disappointing result for United after they dominated the opening period, with Solskjaer admitting they lost their way in a "sloppy" second half. PHOTO: AFP
Pogba wasted a chance to give United their second successive Premier League win when his penalty was saved by Wolves keeper Rui Patricio.
It was the fourth missed penalty by Pogba since the start of last season. The France midfielder appeared to tell Rashford that he wanted to take it, even though his United team-mate had scored from the spot against Chelsea last weekend.
United manager Solskjaer said he allows both Pogba and Rashford to take spot kicks.
"The two of them are designated the penalty shooters and it's up to them there and then who feels 'this is mine'," he said.
"Sometimes players just feel they are confident enough to score. Paul has scored so many penalties for us and today Rui Patricio made a good save.
"The two of them have been very confident. I like players with confidence and the feeling that 'I can do this'."
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Former United star Gary Neville called on Solskjaer to resolve the issue before his team's next penalty.
"They should decide in the dressing room who is the penalty taker - it's embarrassing," Neville told Sky Sports.
"This is a Manchester United penalty, this not a tombola, this is not under-fives on the school field.
"Rashford scored last week - take the penalty. But there wasn't a leader out there. Something wasn't right."
Rashford revealed Pogba had said he wanted to take the kick and, publicly at least, the England striker had no problem with that.
"Paul wanted to take it, it's that simple. Everybody can miss a penalty. He's scored countless penalties for us so it's normal to miss one.
"I took one last week so for me it's no problem for him to take one this week.
"It's unfortunate he didn't score but it's not on him, it's as a team and we take it forward to the next game."
It was a disappointing result for United after they dominated the opening period, with Solskjaer admitting they lost their way in a "sloppy" second half.
"First half was a mature performance. Second half was a bit sloppy. We are improving. We are a young team who will learn, we learned on the pitch today," Solskjaer said.
"Wolves probably got the goal they deserve. I thought we took control again and could have won it.
"We get a penalty that normally would get us another two points, sometimes the keeper makes a good save."
Solskjaer also refused to rule out Chile forward Alexis Sanchez leaving United before the close of the European transfer window.
Sanchez has been linked with a move to Inter Milan and was missing from United's matchday squad at Molineux.
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"Well, there's still a couple of weeks left of the transfer window and there is a chance," Solskjaer said.
"Some clubs have shown interest in Alexis, so we'll see what happens. I can't update you more than that.
"There are a few weeks for international transfers to go through but at the moment he's our player."
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