Cohen has recently led a resurgence of digital channel BBC Three, with viewer numbers up by 56% among the channel's target 16- to 34-year-old audience since he took over in May 2007.
BBC Three was also named 'Non-Terrestrial Channel Of The Year' at last month's Edinburgh TV Festival. Click here to read more about Cohen's session at the festival.
After transforming BBC Three, Cohen has been linked with the new chief creative officer (CCO) role at Channel 4, where he previously worked for seven years.
However, he told The Guardian: "I'm definitely staying at the BBC. I've been here three-and-a-bit years and it's my first experience of the BBC.
"I've never worked here before and I'm really happy here. I think it's an amazing place and I'm definitely staying."
There is also speculation that Cohen could move over to run BBC One should the channel's current controller Jay Hunt secure the CCO post at Channel 4.
In response, he said: "I can't really say any more than I said at Edinburgh, which is that I love running BBC Three and I'm really proud to run BBC Three. I shouldn't get involved in speculation about something like that."
Julian Bellamy, who Cohen replaced at BBC Three, is currently fulfilling Channel 4's CCO role while interviews are being conducted. Bellamy is also a leading candidate to take on the position permanently.