The 36-year-old ex-Man Utd, Man City, Blackburn and Portsmouth star will leave Sunderland after his 12 month deal at the Stadium of Light ends this summer.
Norwich have lost two senior figures in Darren Huckerby and Dion Dublin, and Glenn Roeder is tempted to take a gamble on the striker.
Cole had a three month loan spell at Burnley towards the end of last season, and scored six goals in 13 appearances during his time in Lancashire, including a hat-trick against QPR at Loftus Road in February - his first in seven-and-a-half years.
Cole is still the second in the list of Premier League's all-time leading marksmen behind Alan Shearer.