Following Juve's Champions League exit at the hands of Lyon last week, Maurizio Sarri was sacked as head coach, with Pirlo being announced as his replacement shortly afterwards.
The World Cup winner had previously been put in charge of the reserve side, but was promoted to first team boss after just nine days.
Pirlo is set to undertake a squad evaluation ahead of the new Serie A season, and Toni, a former team-mate, believes Pogba will be on the new head coach's wishlist.
"Knowing Andrea, he will start building from midfield," Toni, also a World Cup winner with Italy, told Tuttosport.
"The ideal gift would be Pogba. I don't know if that's possible, but it would certainly be the best option.
"Andrea will not be a coach who focuses on defending and counter-attacking. He loves to dominate the ball and will want to impose that on his team.
"I don't know what system he will use and I don't think it matters. Andrea is intelligent and flexible."
Although Pogba has become more settled at Manchester United since the turn of the year, prior to that it was strongly rumoured that he had asked to leave the club for a new challenge.
The Frenchman is currently in the last year of his contract with the Old Trafford side, but United have the opportunity to automatically extend it by one year should they desire.
Pogba and Pirlo played together in central midfield during a highly successful period for Juventus a few years ago.