General aviation Flight Instructor with student reported a near miss with another aircraft during approach. ATC issued an immediate climb to the instructor's aircraft when the other aircraft did not maneuver away.

Narrative

Aircraft X; which we were flying; at the time of the event cleared by ATC on the RNAV 09 the AXH at 2000'; was asked if they had traffic in sight to the left; as we were in VFR conditions with my student under the hood for training. I confirmed we had the traffic in sight. We were at an almost 90* right angle track from said traffic with VFR right-of-way conditions; even without our required ALT clearance held. The other aircraft made no heading or altitude adjustments and maintained a direct intercepting path with our aircraft. Their lack of avoidance; by the PIC instructing or the student; forced ATC; Houston Approach; to clear us for an immediate climb so that we could take evasive action and avoid a midair collision with the other aircraft. Upon returning after the flight; I was informed by another experienced instructor that they also had to take immediate action in VFR conditions to avoid the same aircraft. Upon review of the audio recording the tower controller stated clearly to the other aircraft; with plenty of time to change course slightly or altitude; 'If you intend to maintain your current heading you have traffic three miles ahead.' The student responded that they had no contact and almost a minute later responded again with; 'traffic in sight.' They still made no changes to avoid collision; and we were forced to evade in response.