The pair were on Wearside on Tuesday undergoing medicals, which 28-year-old Finland international Tainio, Sunderland's first close season purchase, passed.
Sunderland manager Roy Keane is also targeting two other Tottenham players - Steed Malbranque and Younes Kaboul - the Black Cats boss met the pair to discuss a move to the north east over the weekend.
Keane told the club's website: "I'm delighted to sign a player of Teemu's experience and quality, he'll be a valuable addition to our squad.
"Teemu has showed versatility, playing in different midfield positions for both Tottenham and his previous club, Auxerre.
"But his experience in particular and the fact he has played more than 40 international games shows he has a lot to offer Sunderland."
Tainio played 83 matches in all for Spurs after he signed from French outfit Auxerre scoring three times.