King, 31, has been hampered with chronic knee problems during his career, and the centre-back combats the issue by barely training with the Tottenham first-team squad.
He has managed 21 Premier League starts this season, but has missed Spurs' last three matches, after enduring yet further fitness issues.
King's current deal at White Hart Lane expires this summer, and manager Redknapp is hopeful the stopper will be rewarded for his loyalty as he continues to strive to be available for the side.
Redknapp told The Times: "For the sake of the club and the team, he has tried to play on, but he has been playing in pain and his movement's not been as good.
"Hopefully, we'll get his knee cleared out in the summer and he will be right again. I'm sure the chairman (Daniel Levy) respects everything he has done for the club and we'd love to keep him around."