According to the Telegraph's Chris Bascombe, the Blaugrana are set to bring the winger back to the Camp Nou for £10.5 million.
The La Masia graduate spent the latter half of last season on loan at AC Milan, where he put in an impressive spell, but he has "failed to convince manager Ronald Koeman he should be part of his long-term plans."
Deulofeu had an excellent campaign prior to Koeman's arrival at Goodison Park, notching 13 assists and four goals in the 2015-16 season, but he struggled to make a mark under the Dutchman last year.
He did well for the Rossoneri, though, grabbing four goals and three assists in 17 Serie A matches.
Matteo Bonetti of beIN Sports and football journalist David Amoyal hailed his impact at the San Siro:
According to Bascombe, new Barca manager Ernesto Valverde wants Deulofeu to have a "fresh start" at the Camp Nou.
He wouldn't be the first Blaugrana player to come through their youth ranks, leave and then return, with the likes of Gerard Pique, Cesc Fabregas and Denis Suarez having done the same.
However, Amoyal suggested Barca could sell him for a profit immediately after re-signing him:
The club could use more cover for the likes of Lionel Messi and Neymar, with Arda Turan perhaps still to convince in Catalunya.
However, they could equally do with more money to fund the purchase of players who address more pressing needs in the starting XI.
One such player is Paris Saint-Germain's Marco Verratti, who is reportedly trying to engineer a move to the club, per La Gazzetta dello Sport (h/t Sport). It's said the Italian has told his club, "I won't return to Paris, I want to go to Barcelona."
Football journalist Rafael Hernandez noted Barcelona's need to sign him but also the difficulty they'll face in doing so—and their need to generate funds to bring him in:
Indeed, according to Gazzetta (h/t Sport), PSG responded to Verratti emphatically, reminding the midfielder "We don't sell."
Barcelona's midfield is in dire need of improvement, and Verratti appears tailor-made to help them re-establish their ability to dominate other sides from the centre, but PSG are in a strong position.
The transfer may yet go through if Verratti continues to push for the move and Barcelona are able to piece together a package Le Parisiens might accept, but getting it over the line will be difficult.