WASHINGTON—House Republicans have introduced legislation to lock in cuts to individual tax rates beyond 2025, a proposal that will have trouble advancing in the Senate but which sends a signal about GOP priorities ahead of competitive midterm elections.

The sweeping, $1.5 trillion tax cut enacted into law last year permanently lowered the corporate tax rate to 21% from 35%. But reductions in individual tax rates were scheduled to expire after eight years so the GOP could meet a restriction imposed as a condition of the fast-track...