The 34-year-old arrived at White Hart Lane on a free transfer from Blackburn Rovers last month and, despite making only a handful of appearances for the club, has seemingly done enough to warrant a new deal.
Nelsen scored his first goal for the club during Tuesday night's FA Cup win over Bolton Wanderers to help Tottenham progress to the semi-finals of the competition where Chelsea await at Wembley.
And Redknapp has compared the influence of the veteran defender to that of Scott Parker, who has been pivotal in the club's push up the table.
"I thought he was fantastic tonight," said Redknapp. "I just think he is a great professional and a real good player.
"I couldn't believe I got him on a free. I felt it was an amazing player to bring to the club, he is a player I have chased for years.
"I have always thought he was top class, a great pro, a great trainer, a fantastic person.
"He has got a six-month contract. At the end of the season you have got to keep him here next year because he's worth his weight in gold just to have around for other people to look at the way he works, the way he trains, the way he behaves, his attitude.
"That's what football clubs are built upon, people like him, Scott Parker, Brad Friedel. You bring them in and it changes everything. That has been the difference this year."
Goals from Nelsen, Gareth Bale and Louis Saha was enough to see Spurs progress at the expense of the Trotters, who set up a nervy finish following a 90th-minute Kevin Davies effort.
