Black Cats boss Steve Bruce has publicly ruled out a move for the Wigan midfielder because the deal became "too rich" for the club, but he would dearly like to land a player he has tracked for some time.
N'Zogbia's proposed move to Aston Villa stalled this week when the Latics rejected their offer for the Frenchman, and it remains to be seen whether or not any of the clubs chasing his signature will meet both Wigan's asking price and the winger's wage demands.
However, Bramble is confident the former Le Havre man would welcome a switch to the Stadium of Light.
He told the Journal: "I have spoken to Charles a few times this summer. When I spoke to him last year, he said it would be a fantastic move for him.
"Whether it materialises for him or not, I don't know - but you never know in football.
"I think the players would love him here, the fans would love him to be here - but it's out of our hands.
"We want as many fantastic players as we can here. He's a fantastic player and he would be great for this club."