The French midfielder has been persistently linked with a move away from north London, with Real Madrid and Manchester United among those said to be keen to prise him away from Highbury.
The 26-year-old was quoted on Arsenal's website as telling France Football magazine: "I have entered negotiations to extend my contract, which expires in 2004.
"I am taking part in talks to find a common ground and I cannot see any reason why I shouldn't stay."
"I have experienced so many things with that club, and for me winning the Champions League with Arsenal would be more meaningful than with Manchester United or Real Madrid.
"That is how I see things. I started from nothing and winning the most beautiful trophy with my club would be extraordinary.
"Being the captain of Arsenal fills me with pride." Vieira also believes Arsene Wenger will extend his stay at Highbury.
"He will never leave," said the midfielder. "If he left, the club would be totally different, too many things would go with him."
Wenger is confident Vieira and Thierry Henry will agree to a new deal despite the fact that Arsenal have a wage limit that they are not prepared to break.
"They are well paid already but we will respect our wages structure," said Wenger.
"I think they will make a sacrifice.
"When players express their desire to stay, the financial issue is a strong one.
"They are world-class players and you must pay them as world-class players. It's a minor problem though."
Meanwhile, the Football Association has set a date for Vieira's miconduct charge disciplinary hearing.
Vieira allegedly abused a referee after he was sent-off during the Premiership match with Chelsea on 1 September.