The midfielder is reported to have been the subject of several offers from Championship rivals Nottingham Forest.
Boro have fended off that interest and Mowbray is adamant that the 27-year-old is happy at the Riverside Stadium.
He said in the Middlesbrough Gazette: "The positive for me on Grant is that he's said to me many times he enjoys being at Middlesbrough FC.
"He enjoys the training, he enjoys coming to work every day and I am very confident in my own mind that he's not looking to move on.
"He's very settled and looking forward to the season.
"He was very important to us last season and he will be moving forward with us I'm pretty sure.
"He's happy with Dean Whitehead's arrival at the club, somebody he knows well, and in training they've been bouncing off each other.
"They are both very competitive and it's brought an edge to training. And we're looking forward to the midfield being a strong department this season.
"If anything it's a positive for us that people are interested in our players because it suggests we've got some good players."