Boro have reportedly made three offers for the Scottish striker, all of which have been rejected by Leeds United.
Discussing his side's chances of being able to convince the Yorkshire outfit to eventually part with the 27-year-old, Mowbray has admitted that he does not hold out too much hope.
"It appears to me that every week Ross McCormack seems to score another goal and it might be difficult to prise him away from Leeds but we have tried," he told reporters.
"I don't think Leeds want to sell him. It has leaked out that we had an interest and that will be there if Leeds want to listen, but if not we will get on with our team-building."
Mowbray is believed to have turned his attention to Norwich City's Luciano Becchio.