The 23-year-old is believed to have been shortlisted by QPR manager Harry Redknapp as a replacement for striker Loic Remy after the Frenchman completed a £10.5million move to Stamford Bridge on Sunday.
Italy striker Borini spent last term on loan with Sunderland, notching seven goals in a productive spell and leaving Poyet keen to secure the 23-year-old's services on a repeat temporary deal.
Former Uruguay midfielder Poyet said after Sunderland's 1-0 loss at QPR on Saturday that he remained hopeful of securing a deal for Borini before Monday night's transfer deadline.
Rangers boss Redknapp has moved in with an offer for a permanent transfer that Liverpool have accepted, with a fee that could rise to £13million with add-on clauses.
Redknapp remains hopeful of completing a deal to bring midfielder Sandro to Loftus Road from Spurs, while Jermain Defoe has been heavily linked with joining QPR from Toronto.
Follow Deadline Day on Sky Sports News HQ for all the latest news as clock ticks down toward the 11pm deadline.