Democrats enjoy a strong edge with voters on health care, with 40 percent trusting them on the issue, compared with 23 percent who trust Republicans more, according to an April poll by The Associated Press and the NORC Center for Public Affairs Research.

Forty-two percent of Americans in that survey favored a single-payer system along the lines of what Mr. Sanders is proposing, while 31 percent opposed it and 25 percent did not take a position. Views of the Affordable Care Act are split right down the middle, with 50 percent approving of it and 48 percent disapproving of it in an April poll by Gallup.

Given the Democratic advantage, many Republicans say they should not focus their energy on health care but instead emphasize immigration and other issues where they are stronger. But the president and his team counter that even if they cannot win on health care, it would be ridiculous to simply cede the territory if they could at least narrow the gap.

The president feels compelled to have something specific to counter Medicare for all. Still, people close to Mr. Trump suggested that if he did release something within a month or two, it might not be a comprehensive plan but a series of smaller proposals that would potentially help, such as ideas for bringing down prescription drug costs, for giving states more flexibility on Medicaid expansion and for promoting more access in the marketplace.

Among those who are said to be influencing the discussion within the administration are Alex M. Azar II, the secretary of health and human services; Seema Verma, the head of the agency that runs Medicare and Medicaid; and Joe Grogan, the director of the White House Domestic Policy Council.

“The president has repeatedly promised something better than the A.C.A. but has never come up with a plan himself, and the congressional plans he endorsed were definitely not better for everyone,” said Larry Levitt, a senior vice president at the Kaiser Family Foundation.

“There’s always a tension for presidents around whether to submit a specific proposal to Congress or let the legislative process play out,” Mr. Levitt added. “When it comes to health care, the challenge has been that the president has not only avoided proposing a specific plan, but has made promises that no plan could ever fulfill.”