The two men have developed a rivalry that has continued into the 2017-18 campaign, peaking when Conte described his opposite number as "a little man" and "fake".
"With Jose we have clarified the situation, that's the most important thing," he told reporters. "There is not a problem between him and I. This final is a game between two great teams, Manchester United and Chelsea.
"For sure in my mind I can tell you that we played last season as favourites, I think this season you arrive not as favourites. We are talking about two managers with strong characters, two winners.
"In your blood and heart there is a will to win. We both want to win this trophy. I have great respect for United's story, for Mourinho's story and they have the same respect for ours."
Conte hinted earlier this week that Mourinho had an easier time of things than him during his first stint as manager at Stamford Bridge.