Earlier, one Republican senator said he is "very pessimistic" that the health care measure will be approved, while another warned leaders about retaliation by conservative voters should they react to a collapse of the measure by striking a deal with Democrats.
The timeline for a vote on the debt limit comes after McConnell announced that the Senate's August recess would be delayed by two weeks, which would give lawmakers more time to work on important legislative items, including repealing and replacing Obamacare.
The Hill reports that "Senate Republicans are scheduled to get updated on legislation repealing and replacing ObamaCare during a closed-door caucus meeting on Tuesday".
It's unclear whether House GOP leaders will reciprocate the Senate's schedule change.
U.S. Senate Republicans said Tuesday they will seek to bring their health-care overhaul to the Senate floor next week after a lengthy intraparty struggle, but it remained unclear whether they had the votes to pass the measure or even what form it would finally take.
But McConnell, a seasoned tactician, was forced to postpone a June vote on the bill when he realised he did not have the votes for passage, amid a revolt by some conservative and moderate Republicans.
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"We should be working with our Democratic colleagues", Collins told CBS News.
"If my side is unable to agree on an adequate replacement, then some kind of action with regard to the private health insurance market must occur", McConnell said Thursday at a Rotary Club lunch in his home state.
McConnell said he expects to move forward with the bill next week once the Congressional Budget Office releases its analysis. Charles Grassley, R-Iowa, said on the Fox News Channel of the chances a bill will pass.
Daines participated in a press conference today urging Leader McConnell to cancel or truncate the planned August recess. The CBO score of the first version found that the BCRA would leave 22 million more people uninsured by 2026. A group of 10 GOP lawmakers led by Sen.
"If, by the end of this week, a Senate ACA reform compromise does not look like it is getting closer to success, the Senate effort - and by extension the entire ACA reform effort - may fall apart", Haines said.