The 30-year-old left back left Old Trafford for the Bernabeu in 2007 and has since been a first choice in the position under both Bernd Schuster and current coach Juande Ramos.
However, with his contract due to expire in 2010, Real are considering cashing in on the player's value this summer before he becomes eligible to start discussing a free transfer with other clubs, according to a report in the Mirror newspaper.
The fee of around £4 million and wages in the region of £60,000-a-week would prove little obstacle to Tottenham, though PSV Eindhoven are also believed to be taking an interest in the experienced international.
Heinze has made 19 appearances for Real this season, scoring two goals in the process.
When Heinze left Manchester United, Liverpool manager Rafa Benitez sought to sign him but Sir Alex Ferguson was reluctant to allow the player to join a Premier league rival.
However, Redknapp's interest now paves the way for a return to England for a defender who was a popular figure among United supporters before the emergence of current left back Patrice Evra.
Meanwhile, Tottenham have revealed the latest plans for their new stadium – including the possibility of an ice rink outside the ground.
Spurs want to include a public square as part of the Northumberland Development Project, which would have space for community events such as a temporary rink or street markets.
The new plans also include a 150-room hotel and, inside the 58,000-capacity stadium, a design to seat spectators closer to the pitch.