The former Bristol City, Fulham and Reading man sees his current deal expire in the summer and as yet there have been no discussions over a new deal.
But the confident 28-year-old, who has made 28 appearances for Steve Bruce's side this season, is keen to extend his stay and play his part in helping the Tigers secure promotion to the Premier League.
"I'm not in a position to beg for a contract. I'll let my football do the talking," Rosenior said.
"I've got to do my best for the club, help them get promoted and, hopefully, I'll be a Premier League player next season.
"I'm not being arrogant, but for two years I couldn't see a better right-back in the league. I know I've got the right qualities.
"I'd like to stay and see the club do well and, besides, my nan would kill me if I did leave. That's another big reason to stay."