The Red Devils have been heavily linked with a move for the Juventus hitman in recent weeks, with Tottenham also tracking Dybala.
However, according to Sky Italy, Manchester United have agreed a deal to land Dybala.
They say United will send Romelu Lukaku to Juventus in the swap deal.
The deal is agreed in principle, however, Sky Italy say the transfer depends on Dybala.
If the Argentina international green light’s the move – Ole Gunnar Solskjaer will get his man.
United boss Solskjaer was asked about a possible move for Dybala on Tuesday evening, with the Norwegian refusing to rule out a deal.
“I’m not here to talk about any rumours or speculation about other team’s players,” he said.
“But of course, we’re working on one or two cases, as I’ve said before.
“There’s another ten days before we start the league and hopefully we can announce a fresh face or two.”
The deal could see Lukaku’s time at Old Trafford come to an end.
Inter Milan were set to land the Belgium international, however, the clubs couldn’t agree a fee.
Instead, it looks like Juventus will beat their Serie A rivals to the striker.
Reports on Tuesday claimed Dybala’s agent has flown to London to discuss terms with United.
A lot has been made of United’s interest in Dybala, with ESPN FC pundit Shaka Hislop claiming the striker would be a hit at Old Trafford.
“Absolutely [United should do the swap deal],” Hislop said. “I would do this in the morning.
“I like Dybala a lot. I think he’s an incredible finisher.
“He hasn’t had the best of times at Juventus and has seen himself deemed surplus to requirements.
“But as a finisher, when he’s on form and playing with confidence, I don’t think there are many in world football better than him.
“Given the way Manchester United play, or given the way others play against Manchester United, where you do have possession very close to the box, where maybe Lukaku’s own strength is running in behind defenders with open space, he doesn’t have that many opportunities.
“[So] Dybala is a far better fit. I would do this as quickly as anybody would offer it.”