Neither is the national front-runner, but Mr. Sanders is believed to have gained significant ground in Iowa, and both are on the minds of the president and his team.

Most of Mr. Trump’s advisers see his biggest looming threat as Vice President Joseph R. Biden Jr., who has maintained a steady level of support despite an onslaught of attacks from the president and his team. But there is often a divide between how Mr. Trump and his aides view opponents.

Over the past few weeks, aides to the president in a series of conversations have discussed how to keep the focus on Mr. Sanders, the self-described democratic socialist who favors a significant expansion of government health programs and who is currently at the top of some Iowa polls. They see attacking him as a way to excite his base and draw attention away from other Democrats.

Such efforts tend to be haphazard and revolve around Mr. Trump’s comments at rallies and on his Twitter feed, as does much of the messaging of his campaign. The president, his advisers say, has been in need of a clear target for months, and he believes he is actually hurting Mr. Sanders.

Mr. Trump’s advisers do not necessarily share that view. But they find utility in trying to elevate Mr. Sanders, and aides are discussing ways to keep attention on Mr. Sanders in the lead-up to the Iowa caucuses on Feb. 3.