Bank of America Corp. is preparing to join the premium rewards card fray, planning a new twist on hot products like the Sapphire Reserve card from J.P. Morgan Chase & Co.

Over the past year, rewards cards such as Sapphire Reserve and American Express Co.’s Platinum card have become much sought-after among millennials and big spenders. Cardholders can rack up more rewards with these cards, which they can use to cover travel and other expenses. The cards’ popularity has prompted new entrants and ever-higher rewards—and...