On Capitol Hill, some Republicans are hoping Mr. Mulvaney and others will change the president’s mind on far bigger targets and convince him that structural changes to Social Security, Medicare and Medicaid — the biggest drivers of deficits that are projected to rise over the next decade — are needed to control the national debt and to preserve the programs without substantial tax increases.

Social Security, Medicare and Medicaid consumed nearly $1.9 trillion of the government’s $3.9 trillion in spending in 2016, according to the Congressional Budget Office, and with the number of people 65 and over projected to rise by a third over the next decade, Social Security and Medicare spending is projected to increase from a third of all spending in 2017 to 42 percent by 2027. Including Medicaid and military and federal civilian retirement benefits, federal spending on older adults will rise from 37 percent of outlays in 2017 to 45 percent in 2027 if nothing is done to change the programs.

Even some liberal economists say that will amount to a transfer of funds from poor children and families toward better-off older Americans, because the budget office projects that discretionary spending — where most programs for poor families come from — will be squeezed from 6.3 percent of the economy now to 5.3 percent in 2027, the smallest level since 1962.

With those numbers on their side, Republicans are most likely to use their power in both the executive and legislative branches to push through large cuts to federal programs for poor and working-class Americans, say Democrats and liberal policy analysts — if Mr. Trump eases up on his promises.

“This is the greatest threat to low-income people in my lifetime,” said Olivia Golden, executive director of the Center for Law and Social Policy, a nonprofit organization focused on low-income Americans.

House Republican allies see no real contradiction in Mr. Trump’s campaign promises and what they say he must now do. Since President George W. Bush’s failed 2005 effort to partially privatize Social Security, Republicans have assured retirees and those nearing retirement that any changes or cuts to entitlement programs for older adults would not affect them.