If one is “in the middle of converting” one would be enrolled in RCIA.
If one is seriously considering converting, one should meet with the priest asap. A serious discussion with one’s loved ones or a very close friend to announce your intentions is also prudent. That way, God forbid, should you pass before your instruction and entry into the Church was completed, many people would know what you would need/want for your funeral, which would be certainly held in the Church, as noted in the above post.
Be at peace, and be assured of our prayers.