Student pilot reported ATC contacted the student regarding an NMAC while performing practice takeoffs and landings at a towered airport.

Narrative

This was a student solo flight in pattern at ZZZ. I requested takeoff XXR; ATC instructed me to standby (I was #7 position). ATC then instructed me to hold-short XXR; then cleared me to takeoff XXR. After takeoff on XXR and while on upwind leg (flying best rate of climb); I contacted ATC and requested left-closed traffic (for XXL). In my left turn; I noticed another aircraft passing underneath traveling from right to left. The other aircraft (Cessna) contacted ATC and indicated near miss with my aircraft. At approximately the beginning of my cross-wind I spotted the Cessna off my left shoulder (it seems to me the Cessna was approaching approximately from the northwest direction and below me). ATC then asked where I came from and I responded XXR. ATC indicated they had me down for right-downwind departure and I responded with negative. I proceeded to fly three (3) touch-and-go's and then landed on XXL. ATC Ground instructed me of possible pilot deviation and to contact ATC and provided a phone number to call. After parking and securing the aircraft I contacted ATC by phone at XA:49am local time and spoke with ATC about the situation with my Flight Instructor on the line.While I didn't have clear approval for the left turn; based on the way the second aircraft flew beneath me; it didn't matter which direction I turned (left; right or straight); the aircraft separation would have been the same or less. From a see-and-avoid perspective; based on the direction and altitude the other aircraft flew beneath me; and due my aircraft being a low wing; it was not possible for me to see the other aircraft.Prior to takeoff while I was performing run-up procedure; it was very busy with many aircraft and more controllers coming on line and ATC split the frequency (XXR and XXL).While I should have informed tower that I wanted left-closed traffic before takeoff; it would have had no affect of the relative positions of the two aircraft. Upon calling ATC by phone; they confirmed that the Controller did not provide a direction of flight after departure.