The long serving defender has 18 months to run on his present deal but Spurs are keen for him to sign as he begins to show the kind of form that made him one of the top performers in The Premiership before a long term injury ruled him out last year.
Several clubs have been linked with a move for the 27-year-old, including Newcastle United and North London neighbours Arsenal, and whispers have been circulating that the defender will see out his contract at White Hart Lane and move to The Gunners on a Bosman free transfer, following in the footsteps of Sol Campbell who made a similar move two seasons ago.
Carr has been made captain in the absence of injury-plagued midfielder Jamie Redknapp and has helped the club reach the quarter final stages of the Carling Cup, as well as aiding their good form in the league.
Speaking on Sky Sports News, Spurs chairman Daniel Levy confirmed that the club want to tie their man to a new deal.
"The answer with Stepehen Carr is that we would love for him to stay," said the Spurs chief.
"Every player has a right to see out there contract if they want to leave.
"But we will obviously have discussions with him throughout the year."
In other news, Levy assured shareholders at Thursday's annual meeting that the club would definitely not be joining Arsenal at their new Ashburton Grove stadium when it is built, but did not rule out sharing Wembley with their old foes if their plans for a new stadium do not bear fruit.
If Arsenal fail to complete the project, it has been suggested that both the Gunners and Spurs might share the new Wembley.
The only other option for Spurs if they were to quit their current home, would be to hope London wins the 2012 Olympic Games, and then they could take over the stadium afterwards.
Tottenham's preferred option is to remain at White Hart Lane, but they know they must increase the current 36,000 capacity, and they have secured a £75million bond which would enable them to improve it to 48,000.
However, they will only stay in Tottenham if the transport links are greatly improved and the area is regenerated.
Levy said: "With regards to the stadium it is very difficult in a meeting to explain all the complications that go on with regards to planning applications.
"We are spending lots of money on fees but we may not stay here, we may not enlarge the stadium, and we may not get the necessary approval from the local authority - we have to speculate to accumulate. We want to stay here but this area needs regenerating.
"The transportation issue is much bigger than people think it is, because if we just said we'd go ahead irrespective of whether or not the transfer links would be improved, the council then wouldn't give us planning permission.
"The council will only be comfortable with us enlarging the stadium if there is the transport in and out of the area, so we cannot ignore that issue.
"The club has to improve the first team, the academy and the stadium, and it is a question of prioritising. We still hope to enlarge the stadium and it is a priority but not a top priority, because our top priority has got to be to get a winning side."
Meanwhile, Spurs have offered youngster Mark Hughes a new one-year deal.
The Northern Ireland Under-21 international has featured for the reserves this season and the club are keen to tie him down to a new deal.
