The 31-year-old has failed to receive a Three Lions call-up so far in his career despite currently being the highest scoring Englishman in the Premier League with 14 goals.
However, Lambert is hoping to play for the All Whites by virtue of the fact that his grandfather was born in Christchurch before moving to England after the end of World War II.
"I've made no secret of the fact that I would love to play international football, but I haven't yet managed to break into the England squad," Lambert told the club's official website. "New Zealand has always had a special place in my heart given that my grandfather was born there.
"I hadn't really considered the possibility of being able to play for them until a couple of months ago, when it came up in conversation with Graeme [Staddon]. He said he thought I had a case, so it's all gone from there.
"Everything I've heard from their footballing authorities and their foreign office has been encouraging, so I'm really hopeful that this can get sorted out sooner rather than later. I've been keeping a close eye on how they've been getting on in their recent matches, and hopefully I'll get the call to play for them in the near future."
Lambert scored the winner during Saints' 2-1 win over Chelsea on Saturday.