The Hammers will move from Upton Park to the Stratford-based stadium in time for the 2016-17 season and as a boyhood fan of the club, Tomkins is hoping to earn a new deal in order to play at the 54,000 seater.
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"My contract expires just before it happens but hopefully I will still be around because, just like all the fans, I am looking forward to that occasion.
"This has been my club since I was eight. It's hard to see me ever being anywhere else because I have been here so long, and my family are as excited as I am.
"To see their son potentially playing there makes them very proud. And being a local boy, I will be doubly proud to be there."
Tomkins hopes manager Sam Allardyce will also extend his term ahead of the expiry of his current contract at the end of the season.
"He is doing a great job here," Tomkins said of the 58-year-old. "He got us promoted at the first time of asking and you can't ask for much more, really."