PA-44 flight instructor on training flight with student reported a hydraulic leak inflight.

Narrative

While conducting pattern work at ZZZ; I got a Hydraulic Pump On CAS message. After a few more seconds the message went away. However; I still took flight controls and I cycled the landing gear; afterward the right main landing gear came back with an unsafe indication in addition to a Gear System and Hydraulic Pump On CAS message. My student and I departed the pattern and went out to the practice area to try and solve the issue further. I cycled the landing gear one more time; and the left main landing gear came back unsafe as well. I conducted the emergency landing gear extension procedure and got a 3 green indication. After I got a 3 green indication; I notified approach of our issues and requested priority handling. I did a low approach over Runway XXR to have tower inspect the landing gear before coming back around for a landing. I landed successfully; without issues; and shutdown on the runway. I believe a loss of hydraulic pressure in the landing gear system during flight caused this event. I had a conversation with one of our mechanics once the aircraft was shut down. He suspected that we lost hydraulic pressure due to a hydraulic fluid leak. That would explain the gear system malfunction in the traffic pattern; as well as the landing gear not extending when trying to cycle the landing gear. Once the airplane was towed into the maintenance hangar; the mechanic opened a panel in the baggage compartment and we saw that hydraulic fluid had leaked. I did not find any leaks during the pre-flight inspection. I would include a bigger emphasis on inspecting the landing gear system; more specifically the hydraulic pump and lines when our airplanes are in for maintenance inspections to potentially catch these issues early. That is a strategy that could mitigate this issue; but not prevent it entirely. Since these are training aircraft; components within the aircraft are likely to fail during its use.