The timing is critical because the Senate is expected to take up its health care overhaul this week, and Republicans — who control the body with a slim 52-vote majority — have already lost the support of two of their senators. Losing one more Republican senator would effectively sink the legislation, and a handful of Republican senators from states that have expanded Medicaid under the Affordable Care Act have signaled they will closely follow the lead of their state’s governor.

At a private luncheon for governors on Saturday, three Democratic governors called for the group to release some sort of joint, bipartisan statement on the health bill. Gov. Dannel P. Malloy of Connecticut proposed a letter formally opposing the Senate legislation. Gov. Terry McAuliffe of Virginia, the chairman of the National Governors Association, and Gov. Steve Bullock of Montana suggested a more restrained approach to communicate unease with the health bill that could garner bipartisan support from the heavily Republican group. But a handful of Republican governors opposed making a collective statement, noting there was no broad agreement about the nature of their opposition.

“It’s important if anything goes out under the name of the N.G.A. that it has the endorsement of members certainly, and I think there was not consensus on that,” Gov. Matt Bevin of Kentucky, a Republican who voiced his discomfort with a joint communiqué during the lunch, said after the session.

Nonetheless, many of the Republican governors appear to be opposed to the current Senate legislation, and there is widespread skepticism in their ranks that the bill will pass.

Gov. Scott Walker, Republican of Wisconsin, who has previously voiced his irritation that the Trump administration did not originally seek the views of governors on health care, said at the luncheon that the best time for the governors to collectively weigh in would be after the Senate bill fails, according to officials in the room.