A320 Captain reported experiencing constant uncommanded pitch up movement while on manual flight during climbout requiring forward pressure. The flight crew contacted Dispatch and Maintenance and decided to return to the departure airport.

Narrative

I was the pilot flying and departed ZZZ Runway XXL. As I began climbing out; I noticed the aircraft required forward pressure on the side stick. Once I leveled off; and was level on the flight directors; the aircraft still required forward pressure to maintain level flight. The aircraft would constantly pitch up. We were given a higher altitude and I climbed to it. All the while having to push slightly forward on the side stick in the climb. I engaged the autopilot and it maintained the climb and then cruise.Once our workload was reduced I contacted dispatch and maintenance on the satellite phone. I told them that when the autopilot was off; the aircraft required forward pressure on the side stick. Both Flight Directors were indicating the same. Also; when the FO flew; he noticed and felt the same problem. Maintenance had me disconnect the autopilot while in level flight and I noticed the airplane again pitch up. We all agreed the safest course of action was a return to ZZZ. Maintenance also noted that as we burned fuel and the CG changed; the aircraft could be harder to fly manually.We notified ATC that we would be returning to ZZZ. We also [requested priority handling] as we had some sort of issue with pitch trim in manual flight. I used the [priority handling] call button and gave the flight attendants a heads up. I told them to expect a normal approach and landing. I let them know they would also see emergency vehicles on our landing roll out.We had a long final for XXL at ZZZ and configured early. As I turned the autopilot off; the aircraft still wanted to pitch up. We landed uneventfully back and ZZZ and taxied to our gate.I entered the issue in the logbook and also spoke with the two mechanics that met our aircraft at the gate.