Instructor pilot reported an NMAC during a training flight when climbing to VFR cruise altitude and receiving VFR traffic advisories from RHV ATC. The pilot was scanning for the conflicting traffic as provided by the ATC advisory while instructing student to increase climb rate above planned level off altitude and turn direction. The conflict aircraft passed 200 ft. off right side of the training flight original planned altitude and was at an incorrect VFR altitude for direction of flight.

Narrative

Myself and my student had departed Reid Hillview Airport to the south east practice area. We were intending to level out at 3500 ft. traveling southeast. Got a traffic advisory from Reid Hillview Tower of a plane at my 1 o'clock in 3 miles; roughly the same altitude and opposite direction. I could not see the traffic and did not know whether they would pass off my left or right; or if they were climbing or descending. After about 10 to 15 seconds of scanning; I told my student to make a turn to the left and begin a steeper climb and we would not be leveling out at 3500 [ft]. At the same time; I got another traffic advisory from Reid Hillview Tower about the same plane off my right side. I responded that I was still looking. Less than 2 seconds later; I spotted the plane pass just off my right side and about 200 ft. below me. This plane was traveling VFR westbound at 3500 ft. (Incorrect VFR altitude for direction of travel). If we had not made that slight left turn and started a steeper climb; we very likely would have collided.