First, I completely agree with balanced mama (I usually do....) about making sure that SAFETY is your first priority. There are a lot of people who delude themselves into thinking that if their child can doggie-paddle, then he/she is a perfectly fine swimmer and can be left unsupervised in a pool. This is not the case. Kids who can doggie paddle are barely swimming. It is not going to harm your child to be overly-cautious with him/her until his/her swimming proficiency progresses adequately.
Maybe now isn't the time for swim lessons, but safe exposure to water is important. Giving your son the opportunity to explore water in a safe environment will make him more comfortable and, eventually, he'll learn how to swim in his own time.
My mom first put me in swim lessons when I was 4 or 5. And I was a screamer. I clung to the wall of that pool and refused to budge. I remember it vividly. For the duration of those swim classes, I probably ventured from the wall a total of 3 times and I'm certain I screamed more than I didn't scream. That summer, I started daycamp and we went swimming EVERYDAY. I didn't learn to swim that summer or the next summer, remaining safely in the shallow end of the pool where I could touch the bottom (I wasn't stupid. I knew I couldn't swim and I knew exactly how far I could go before it was too deep for me).
Finally, the summer I was 7, I decided that I wanted to go off the diving board. My daycamp, understandably, would not allow anyone who couldn't swim the width of the pool to jump off the diving board into 8 feet of water. I watched my friends and kids who were younger than me happily leaping off the end of the board and pop up smiling as they swam to the edge of the pool, and I felt stupid. So, in the span of about an hour, with the help of some daycamp counselors, my friends, and a swim instructor who had been teaching swimming longer than I'd been alive, I learned how to swim.
Over the years, I became a swim instructor myself, I was a lifeguard, and I'm proficient in every stroke and kick (maybe not the butterfly so much, but I could teach it if I had to...). I can tread water for several minutes and rescue unconscious victims from the bottom of a pool. I wouldn't win any awards for diving, but I've taught a fair number of children to dive, too.
Before learning can begin a child really needs to be comfortable in the water and feel motivated to learn how to swim. That motivation is different for every kid, but it needs to be an intrinsic motivation. Begin by getting him comfortable in the water without a life jacket with you there and worry about the lessons later.