The 40-year-old has steered his side to the top of the table in what has been a solid campaign to date.
And, having helped the Whites knock Manchester United out of the FA Cup in recent weeks, his reputation has soared - making him the ideal candidate for the Clarets post.
Despite such rumours, Grayson says he intends on staying with Leeds - a side he deems perfectly capable of making a Premier League return in the not-too-distant future.
"I have only been here a year and it took a lot for me to leave Blackpool. I had a good job there," Grayson told the Yorkshire Post.
"This was a fantastic opportunity and I have said all along that someone will take this club back into the top flight. Hopefully, that will be me."
Elland Road has become somewhat of a fortress for Leeds, who have not lost a league game at home in 12 months.
Grayson admits that the club's passionate following has a large part to play in that success, as it did at Old Trafford just a few days ago.
"I am an ambitious manager but this is an ambitious football club," he said.
"You saw the passion of our fans at Old Trafford last week and then how we got 25,000 in on a bitterly cold day like Saturday (against Wycombe)."
He added: "I am fully focused on achieving my target of getting this club promoted into the Championship and beyond.
"People can speculate with Burnley or whatever club they want but I am committed to trying to take this club forward.
"I am flattered because everyone wants to manage in the Premier League. But my ideal scenario is to manage this football club in the Premier League."
Leeds 4/6, Draw 13/5, Exeter 9/2