The leadership plan was defeated by a procedural vote of 274 to 149.
Instead, they will be taking up a procedural vote required under Senate rules to move forward to the actual debate.
The House, in a 356-to-58 procedural vote, supported the same provision.
The action came in two procedural votes as the House set the stage for considering the overall bill next week.
A procedural vote in the Senate indicated an extension has limited support.
The measure barely survived in a procedural vote to move it to the floor.
On a key procedural vote early in the day, the House adopted a resolution known as a rule, which set the terms for the debate.
But the defeat, on a procedural vote of 58 to 40, by no means put the thorny issue to rest.
The amendment was defeated in a procedural vote of 58 to 36, largely along party lines.
Their effect on the issue suggests that the procedural votes the Senate held this month are unlikely to be the last word.
