The 32-year-old midfielder still has two-and-a-half-years to run on his existing contract at White Hart Lane, after he arrived in north London when Redknapp was in charge in August 2011.
Parker has missed most of the season through injury, but recently made his first-team comeback, under the new management of Spurs boss Andre Villas-Boas.
The Portuguese tactician will have been delighted to see Parker's timely return, after fellow midfielder Sandro was ruled out for the remainder of 2012/13, after suffering a knee injury in Tottenham's goalless draw with QPR earlier this month.
The Brazilian underwent surgery in Barcelona and has been given a minimum four month lay-off by doctors concerned about possible long-term implications to the injury.
As a result, Spurs brought forward the signing of Schalke attacking midfielder Lewis Holtby - who initially signed an agreement to join the Premier League outfit on a Bosman free in the summer - paying £1.5million to push the transfer through this month.
Holtby's arrival has reduced the pressure on Villas-Boas's midfield options, and could now pave way for Parker to leave for QPR.
Redknapp has drawn up a shortlist of targets he wants to sign before Thursday's transfer deadline, including Stoke City striker Peter Crouch and former Blackburn Rovers defender Christopher Samba, now of Russian side Anzhi Makhachkala.