Talks between the two clubs are expected to continue over the weekend, with Di Gregorio having already given his approval to a move that would take him outside Serie A for the first time in his career.
Bournemouth have been weighing up reinforcements in goal throughout the summer, and a deal for the 29-year-old would likely see Will Dennis head out on loan of his own, with the Englishman previously pencilled in as competition for Djordje Petrovic.
Di Gregorio has made 84 appearances for Juventus since joining from Monza in 2024, keeping 36 clean sheets, including five in 14 Champions League matches, but he is expected to fall down the pecking order in Turin following the Bianconeri's move to sign Guglielmo Vicario from Tottenham Hotspur.
Juventus are also working to offload back-up goalkeeper Mattia Perin as part of a wider overhaul of their options between the posts.
If completed, the deal would represent Bournemouth's most senior addition to the goalkeeping department this summer.
The club has also strengthened at academy level with the signing of 17-year-old goalkeeper Adam Brogan, who joins after progressing through Southampton's youth ranks and has represented the Republic of Ireland at youth international level.
Brogan will link up with Bournemouth's Category One academy setup ahead of the new campaign.
"I am delighted to be here and can't wait to get started," Brogan said. "AFC Bournemouth is a club that is on the rise, especially with the academy now being Category One, which means being able to play against top opponents. Everyone at the club has been very friendly and it's been amazing so far. I am really looking forward to the season ahead!"
Sam Gisborne, head of Bournemouth's academy, said the club had monitored Brogan for some time before completing the move.
"We had been tracking Adam for several years because we have competed against him when our age groups have played each other," Gisborne said. "There was an opportunity to bring him in, so we are pleased that he decided to choose us."