One of the costliest years for natural disasters puts some U.S. insurers in a position to do what they haven’t done in years: raise prices.

The biggest rate increases are expected from reinsurance companies that could be asked to pay more than half of all claims attached to a string of Atlantic hurricanes, Mexico earthquakes, California wildfires and winter storms. Reinsurers, which essentially act as insurers for other insurance companies, absorb the risk of some policies sold by primary carriers and typically pay once claims...