General aviation pilot reported a NMAC while entering the pattern for landing. Pilot continued the approach for the touch and go.

Narrative

I had contacted ATC at LEE from the East for authorization into the airspace and clearance for a touch and go and departure to the Northeast. Two-way radio contact with tower was established and I was given instructions to join the right downwind for runway 13. I had established this and another aircraft was taking off runway 13 below me. I recall hearing tower asking the other aircraft if he had visual of me joining the right downwind; but didn't hear a response. I had no visual of the other aircraft below me. I continued to join the right downwind as instructed when I noticed off my right side at approximately 1 O'Clock the other aircraft flying approximately 300 feet below and from my right side. I could be wrong on the distance as I had no TCAS warning on Foreflight. Tower also did not give any instruction on avoidance and after the event; I continued on my approach; performed the touch and go and proceeded to head to the northeast. I may be off on the proximity of the other aircraft; but wanted to file this in case something came up. Thank you.