According to Sky in Italy (h/t Sky Sports), the two clubs have agreed a deal in principle to exchange the players, and the move is now contingent on United agreeing terms with Dybala, whose agent is in London to hold talks with the Red Devils.
Per ESPN FC's Rob Dawson, United travelled to Norway on Monday without Lukaku to face Kristiansund in a friendly on Tuesday, but Solskjaer said the striker's absence was down to his fitness:
"It was best for him [Lukaku] to stay home. He wouldn't have been able to play. Hopefully he might be able to train in the next couple of days. Let's see what happens.
"You never know what happens in football. He's not been able to train, but he's not been that bad that we had to send him home, we still had hope he was going to be fit to play in Australia, Shanghai and Singapore. But for this short trip, there was no point bringing him."
The Belgian has not managed to appear for United in pre-season because of an ankle problem.
It's said Dybala, who featured for Argentina in the Copa America earlier in the summer, will return from his holiday on Thursday and wants to speak to Juve boss Maurizio Sarri before deciding his future.
Juventus vice-president Pavel Nedved would not be drawn on specifics relating to Lukaku or Dybala during a recent discussion with the media:
The Press Association's Simon Peach believes Solskjaer will be happy if the club can pull off a swap deal:
Lukaku, who turned 26 in May, is a much more prolific goalscorer than Dybala. The former has 235 career goals for club and country, while the latter has 104.
Their positions differ, though, as Lukaku is a centre-forward while Dybala has often been used in a support role, while last season he spent a significant amount of time on the right flank.
The Argentinian is also far superior technically and more dynamic in his movement than Lukaku, so he could be a better fit for Solskjaer's plans.
Whichever player United end up with, they'll hope to have them fit and ready for the start of the new season.