On a vote of 219-209, with most Republicans in opposition, the House approved the rules for debating the measure, clearing the way for a vote on passage.
In addition, 139 House Republicans and eight Senate Republicans voted to challenge the results in some key states.
Lawmakers in both parties acknowledge the vague law needs to be updated, but the House version faced strong opposition from House Republicans who argued this bill goes too far.
For instance, the Senate bill would require one-fifth of the House and Senate to object to states' electors, instead of one lawmaker in both chambers.
The House bill, on the other hand, requires one-third of each chamber to issue an objection.