Mitch McConnell has a math problem. The Senate majority leader needs 50 votes to repeal Obamacare—but he has no idea how to get there. Even before the Congressional Budget Office weighed in on the G.O.P.’s divisive replacement plan, which the federal scorekeeper estimated would result in 23 million fewer people having health insurance by 2026 than under the current law, Senate Republicans were signaling that they would be rewriting the legislation on their own. But even if they manage to craft a bill that is less politically toxic than the remarkably unpopular version passed by the House, it is unclear whether the lower chamber would approve—let alone whether the Senate could pass it in the first place.

The deluge of scandals battering the White House certainly gives McConnell few incentives to stick his neck out for the president. On Thursday, NBC News and The Washington Post reported that Jared Kushner, Donald Trump’s powerful son-in-law and senior adviser, has come under scrutiny in the Justice Department probe into ties between the Trump campaign and Russia. The news follows weeks of bad headlines about the president’s interactions with law enforcement and intelligence officials that have raised questions about whether Trump has attempted to obstruct the F.B.I. investigation. And now, with the president officially lawyered up and his supporter base shrinking, Senate Republicans seem less willing than ever to take unnecessary risks.

McConnell is acutely aware of the uphill battle he faces stewarding a massive health-care overhaul through the Senate. “I don’t know how we get to 50 [votes] at the moment,” the Kentucky senator conceded during an interview with Reuters on Wednesday. Senate aides are expected to begin drafting parts of the health-care bill next week during the Memorial Day recess. “Staff will start putting together some language we can look at when we return,” Senator John Cornyn of Texas told The Wall Street Journal, but added, “There is no final agreement yet.” Senator Marco Rubio echoed the sentiment, characterizing the bill as a “work in progress.”

It is not clear, however, how the Senate will address thorny issues like preexisting conditions, and there is a growing sense of frustration among some Republicans with their lack of progress. “We talk about it every damn day,” one G.O.P. senator, speaking on the condition of anonymity, told Politico. “But we haven’t done anything about it.” Others viewed McConnell’s concession that he doesn’t know where the votes will come from as a warning that the health-care overhaul may never happen. “He doesn’t do much that’s not purposeful. So is he sending a message here of: ‘Don’t anybody think this is likely to happen?’” another Republican senator told Politico. “If I had to bet my house, I’d bet we don’t get it done.”

Whether Senate Republicans ultimately choose to abandon health care could hinge on what they encounter when they return home to their constituents over the recess. The Journal reports that numerous advocacy groups are launching targeted campaigns, both in support and in opposition of the House G.O.P. bill. Vulnerable Republican senators like Dean Heller of Nevada and Jeff Flake of Arizona, among others, can expect to be targeted by the A.A.R.P. over the repeal of the Affordable Care Act. The conservative American Action Network, in turn, is launching a $2 million advertising campaign in support of the Republican legislation.

But the Senate appears antsy to move on. Some senators have begun to indicate that they might skip right over health care and jump to tax reform, the other big ticket item on the G.O.P. agenda, if the former becomes too vexing. “Taxes has more consensus with Republicans and some Democrats,” Senator Rob Portman told the New York Times. When asked whether overhauling the tax code might prove a lighter lift, Senator Pat Toomey said, “I allow that’s a possibility.”