Trump has been telephoning members of Congress in recent days urging action on dismantling Obamacare.
McConnell has not promised to bring Graham-Cassidy to the floor for a vote, but he asked CBO for a quick assessment of it.
Not all the Senate’s conservatives back the bill. Senator Rand Paul said it did not go far enough to repeal Obamacare.
Senators John McCain (R-Az.), Lisa Murkowski (R-Alk.) and Susan Collins (R-Maine), all of whom voted against the last Republican effort to repeal Obamacare, were said by aides to be undecided on the latest proposal.
But if it can get through the Senate, it may have a chance in the House of Representatives, also run by Republicans.
Past Republican proposals to dismantle Obamacare have been hampered, in part, by CBO estimates that showed the bills would have left millions more Americans without health insurance.
Obama’s healthcare reform provided health benefits to 20 million Americans. Since its passage, Republicans have sought to undermine it. But the party has failed to repeal Obamacare or enact a replacement system, despite controlling the Senate, the House and the White House since the November 2016 elections.