This may sound like a statement to shrug off and ignore but this is one of the claims that has science on its side. Although the studies are not exhaustive, the evidence points towards avoiding hot tubs when trying for a baby.
During puberty your “balls drop” in order to keep them at a lower temperature, ideally 2.5 degrees celsius below body temperature. By spending long periods of time in a hot environment like a hot tub or bath, you are elevating the temperature of your testicles which can damage the sperm and also hinder sperm production.
In a small study exposing men to water baths of varying temperatures scientists found that sperm function decreased but that this was reversible. The study took place over 16 weeks which is long enough to incorporate an entire sperm cycle. Sperm motility was most affected by the high temperatures, reducing the ability of the sperm to swim when attempting to fertilise an egg. The researchers themselves noted the sample size was small and larger investigations needed to be done, however, this does give reason to avoid wet hot temperatures.
The bottom line for this one is that spending a lot of time in hot water may temporarily damage the sperm currently being held in your testes. However, it shouldn't affect a man's ability to conceive as the damage will only be done to that round of sperm. You might want to avoid hot tubs when trying to get pregnant but no need to deprive yourself the rest of the time!
References: