Pulisic won't actually arrive at Stamford Bridge until the summer, while Maurizio Sarri's relatively late arrival in mid-July left little opportunity for squad assessment and new signings at the time.
The latest from Sky Sports News is that Chelsea are now 'close' to agreeing deals for both Barella, a 21-year-old Italy international also linked with Arsenal, and Paredes, who made his name in Italy with Empoli and Roma before moving to Russia in 2017.
Barella, who has been priced at a reported £45m, seems particularly keen to make the switch to English football, admitting that should he go ahead and leave Cagliari in the near future he would prefer it to be for the Premier League rather than another Serie A side.
"If I had to choose between the Serie A and the Premier League, I would go for England because I love the way they approach the game," he told Gazzetta dello Sport.
"Plus, everyone there is just as irritable and prone to fouls as me!"
Originally Manchester City-bound, Jorginho followed Sarri to London from Napoli, while Mateo Kovacic is on loan from Real Madrid. Sarri also still has Ross Barkley and Ruben Loftus-Cheek at his disposal, with Danny Drinkwater set to fall even further down the pecking order.