American Express Co. is taking another shot at J.P. Morgan Chase & Co.’s popular Sapphire Reserve card, planning to increase the points it is doling out to consumers who sign up for its Platinum card and spend a lot in the first three months.

AmEx in coming days is expected to increase the points bonus to 60,000, from 40,000, according to a person familiar with the matter. But there is likely to be catch: To get the larger amount of points, card users will have to spend $5,000 in the first three months after receiving...