The 25-year-old moved to the Riverside from Fenerbahce in the summer and has featured in seven games for Boro already this season, but rumours had hinted that he was unhappy on Teesside.
However, when asked about the stories, Southgate told Boro's official website: "It's nonsense, so I'm not prepared to be distracted from our preparations for Saturday's game against Sunderland.
"I actually saw these supposed quotes from Tuncay a couple of weeks ago and asked him about them. There's no truth to them. He's committed to the challenge."
"His reaction after the game last Saturday certainly didn't suggest to me someone that wasn't 100 per cent committed to the club and what we're trying to do."
"Tuncay has been a substitute for some of our early games, but there's every chance he'll be playing against Sunderland on Saturday."
"The truth is that we've been delighted with him and I'm sure he'll be a big success over here. He's a smashing character and a great professional."