Crypto companies such as Coinbase Global and Circle have sought legislation to bring regulatory clarity to their businesses after costly battles with regulators under the Biden administration. They found allies in Trump and many Republicans after pouring money into last year’s election.

The House this week took up a trio of bills that would establish a federal framework for regulating the cryptocurrency industry.

One of the bills, which would regulate a type of cryptocurrency called stablecoins, has already passed the Senate with broad bipartisan support and will head to Trump’s desk if it passes the lower cha...