The England international believes Spurs will have a chance at St Mary's on Sunday if they replicate the first hour of their Capital One Cup performance against West Ham
Spurs travel to St Mary's Stadium for their first league match since the departure of manager Andre Villas-Boas, with caretaker boss Tim Sherwood having suffered Capital One Cup defeat to West Ham on his managerial debut at White Hart Lane.
The London club are seventh in the Premier League but the 5-0 loss to Liverpool, which ultimately cost Villas-Boas his job, has cast serious doubt over their top-four hopes.
Victory over Mauricio Pochettino's men would give Tottenham a boost ahead of a busy end-of-year schedule but Walker, who was named captain against West Ham, knows it is a tough ask.
"It will be difficult," he told the club's official website. "Southampton have a lot of quality players and a few have been called into the England squad.
"It will be tough but if we play like we did in the first hour against West Ham, with the same enthusiasm and desire, getting the ball in the box and getting shots off, why can't we go there and win?
"I remember last year we went on a good run at Christmas, so why not do the same this Christmas?
"As I said, Southampton will be difficult opponents but if we play like we did for the first hour against West Ham, we can get a positive result."