CAUTION: SPEED BUMPS AHEAD. Latest Push for a Health-Law Repeal Picks Up Speed in the Senate.

Mr. McConnell said on Tuesday the Senate leadership was “in the process of discussing” whether to hold a vote on the latest bill, which would unwind much of the 2010 Affordable Care Act, known as Obamacare.

“If we were going to go forward, we would have to act before Sept. 30,” Mr. McConnell told reporters. “We are in the process of discussing all of this. Everybody knows that the opportunity expires at the end of the month.” He was referring to the short shelf life of the procedural tool that allows Republicans to pass bills through the Senate with a simple majority.

GOP Senators met Tuesday to discuss the repeal bill from Sens. Lindsey Graham (R., S.C.) and Bill Cassidy (R., La.) and two hearings are planned next week. The White House also threw its muscle behind the legislation, with Vice President Mike Pence saying Tuesday that the administration is “all in” on the effort.

The bill still faces the same challenge that sank a July repeal effort: Moderate Republicans worry the proposal goes too far in rolling back the current health law.

Senator Rand Paul says he won’t vote for the bill because it doesn’t go far enough, which puts the new replacement effort in the exact same bind as the previous replacement effort.