The 28-year-old Bosnia international has become one of the most revered players in his position, and a set-piece specialist, over the last couple of seasons and has caught the attentions of Spain's biggest duo, according to separate reports.
According to Sky Sports (via SPORT) Real Madrid see Pjanic as a potential ideal long-term successor to 32-year-old Luka Modric as they look to freshen up their squad this summer.
Meanwhile, Pjanic's current club Juventus are eyeing a move for Real Madrid's Mateo Kovacic according to La Gazzetta Dello Sport, although there is little suggestion of a potential exchange deal. However, the likely departure of at least Dani Ceballos from Real Madrid means Zinedine Zidane's side are likely to be open to more midfield arrivals.
In addition, French site Onze Mondial claims that Barcelona would also be tempted into the race this summer. That is, despite the £142m outlay for Philippe Coutinho in January.
There is little other than market value speculation in any of the reports regarding Pjanic's potential price tag. However, with the former Roma man under contract until 2021 in Turin, he would be unlikely to go cheap, particularly if the two Spanish clubs were to initiate a bidding war.
Pjanic scored six goals in all competitions for Juventus last season, registering nine assists, as Juventus scooped a seventh straight Scudetto and the Coppa Italia.