It is a debate that has many Democrats fretting in competitive races in the House and Senate, where Democrats are hoping to win back the House and defend 10 Senate incumbents in states Mr. Trump won in 2016. Of the 67 candidates listed by the Democratic Congressional Campaign Committee’s “Red to Blue” list, only one — Randy Bryce in Wisconsin — has been a vocal supporter of abolishing ICE. Many see the issue as a needlessly confrontational framing of a position many Democrats share: reforming, but not eliminating, the agency, and overhauling the way it has operated both under Mr. Trump and former President Obama.

“Democrats are fighting for bipartisan, comprehensive immigration reform that is tough, fair and keeps Americans safe,” said Meredith Kelly, the communications director for the Democratic Congressional Campaign Committee. “Rather than abolishing ICE, we must reform this broken agency to bring more accountability, morality and strength to our border security.”

Other Democrats around the country have been more blunt.

“It’s crazy town,” said Senator Heidi Heitkamp of North Dakota, rolling her eyes when asked about the issue in an interview with The Times last month at a Dairy Queen in Langdon, N.D. The junior senator is facing a tough challenge from Representative Kevin Cramer and the nonpartisan Cook Political Report has the race listed as a toss up.

Ms. Heitkamp said there had been misinformation and misunderstanding surrounding ICE’s role, which can get lost in the emotionally charged debate around immigration.

“I don’t think some people understand that ICE is interior,” said Ms. Heitkamp, meaning that the agency operates within the country’s borders, while Customs and Border Protection is responsible for border security. “And you have to have someone who is going to be responsible for enforcing our immigration laws in the interior, and that’s ICE. And you can disagree with how they’re doing their job, but that’s not their choice. They’re not some rogue agency who’s not following orders over at the Department of Homeland Security.”