Andre Villas-Boas looks set to replace the 33-year-old, but Parker's wages will frighten off the Black Cats.
He is one of the big earners at White Hart Lane, on a wage that would make him the highest paid player at the Stadium Of Light.
The former Newcastle skipper suffered an injury plagued season in Andre Villas-Boas' first year in charge of the club - making only 15 league starts.
And the Spurs manager has indicated that it is a priority to replace the England international over the summer.
Villas-Boas is keen to conclude a deal for Manchester City's Gareth Barry and raise the £4million needed with the sale of Parker.