BALTIMORE—As early Democratic presidential nominating races tighten, many influential Democrats are emphasizing positions that are more aligned with those of Hillary Clinton than Bernie Sanders, even those lawmakers who haven’t endorsed anyone.

Many Democrats think nominating Mrs. Clinton would be better for races further down the ballot. Heading into the first presidential contests in Iowa next week and New Hampshire a week later, Mr. Sanders’s momentum is triggering some concerns among Mrs. Clinton’s supporters on Capitol Hill.

Democrats in Congress like Mr. Sanders, a senator from Vermont, and many are reluctant to criticize a candidate who has been igniting enthusiasm, particularly among younger voters. But in recent days, many have been highlighting themes in line with Mrs. Clinton’s brand of pragmatic politics as opposed to Mr. Sanders’s call for a populist revolution.

Support for Mrs. Clinton, long considered the front-runner, had been building even before President Barack Obama this week came right up to the edge of endorsing her. Almost three-fourths of the Democratic caucuses in each chamber of Congress is now backing the former senator and secretary of state, according to data compiled by Roll Call. Her supporters say she is both more likely to win the White House and would do more to boost Democrats in tight House and Senate races.

“I have profound concerns about his electability in a general election, just as my Republican colleagues have concerns about some of those who lead the GOP pack,” said Rep. Adam Schiff (D., Calif.), who has endorsed Mrs. Clinton. A victory for Mr. Sanders could have “very significant downstream consequences in House and Senate races,” he said. “We want the strongest candidate leading our ticket, and that is Secretary Clinton.”