The Mirror says Redknapp is returning to former club Tottenham for the midfield general, who was one of his best buys for the White Hart lane club.
Parker has had an injury-hit campaign, and Spurs – seeking to strengthen in the middle of the park themselves over the summer – may cash in on the 32-year-old.
Redknapp knows Parker well and will try to persuade QPR owner Tony Fernandes that the experienced England international is an ideal man to both change the culture and improve the playing style at Loftus Road.
Parker could also collect a big final pay-day if he becomes another of the Hoops' high-earning stars.
It would probably mean the end of his England ambitions in a World Cup year however, so this would be a huge decision for him.