Juventus have been pushing to sign Lukaku, 26, this summer and the Italian champions approached United over including Dybala, 25, in their offer for the Belgium international.
Ole Gunnar Solskjaer has made it clear that Lukaku would be playing second fiddle to Marcus Rashford should he stay on at United next season and the striker has pushing for a move to Serie A, with Inter Milan also interested in his signature.
According to Sky Italia, Juve’s talks with United have advanced at a rapid pace and the two clubs have agreed a deal in principle over an exchange involving Lukaku and Dybala.
United must now agree personal terms with Dybala if they wish to finalise a transfer, but the Argentina international is still weighing up his options.
Though Dybala would rather remain at Juventus for the time being, he will make a final decision on his future in the next few days.
His agent and brother, Mariano, has traveled to England this week to hold discussions with the Red Devils over a possible switch.
After United’s pre-season victory over Kristiansund on Tuesday, Solskjaer confirmed that the club are still working on one or two more signings when pressed on rumours of a possible move for Dybala.
Asked how hopeful he was of landing Dybala, Solskjaer replied: ‘I’m not here to talk about any rumours or speculation about other team’s players.
‘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.’
While Lukaku impressed with 27 goals in his debut season at United under Jose Mourinho, he only netted 15 times last term and failed to win Ole Gunnar Solskjaer’s trust, with the Norwegian favouring Rashord as his lead striker.
Inter have also been heavily linked with Lukaku this summer and were reported to have agreed personal terms with the former Everton and Chelsea man.
However, Inter failed with a €60m (£54m) offer for Lukaku and are yet to tempt United with an improved offer this summer.
Asked about the club’s failed bid for Lukaku, Inter CEO Beppe Marotta said: ‘The club is strong, but we need to stay within certain parameters.
‘The club has made big investments and we believe that we’re already at a good stage in the process.
‘We are ready to pounce on opportunities, but prices are nowadays very high indeed.
‘We made a strong offer [for Lukaku], one that is valid and suited to the transfer value of the player and we were unable to conclude a deal.
‘We’ll continue to monitor the situation and anything can happen on the market.’