WASHINGTON—Senate Republicans hit a significant stumbling block Thursday in their efforts to overhaul the tax code, forcing them to contemplate walking back their plan for a permanent reduction of the corporate tax rate to 20% from 35%.

Many congressional Republicans and President Donald Trump set the 20% corporate rate without an expiration date as a fixed pole in their tax policy, and it drove nearly every other decision as they put together a $1.4 trillion plan to reshape the tax system.

But...