Dyche was asked whether the club had moved for O’Brien to replace Jarrad Branthwaite.
The latter is a target for Manchester United, who had two bids rejected this summer.
Dyche told Sky Sports: "No way! You're kidding me! I'm very surprised you've put two and two together at Everton Football Club. It's nothing to do with that. It's just to do with trying to model the club with the finances available. Obviously we're stretched at the moment with depth.
"Trying to bring in players who can add layers to the club. Younger players who can develop and protect the club both on and off the pitch with their performances and then in the future with the finances. It's a tough juggling act and it's tough to find the players who can do that. We think we have a few here.
"We've made it clear to him (Branthwaite) that there's nothing other than that. But of course put two and two together and everyone starts speculating. What do you do?”