President Trump’s advisers see Senator Bernie Sanders as their ideal Democratic opponent in November and have been doing what they can to elevate his profile and bolster his chances of winning the Iowa caucuses, according to Republicans familiar with the plans.

But their new focus on Mr. Sanders, independent of Vermont, comes at a time the president himself has been closely watching Michael R. Bloomberg, a late arrival to the Democratic primary race, unnerved by his campaign spending and his suggestion he might spend $1 billion of his own fortune on opposing Mr. Trump, even if he does not emerge as the nominee.

It has left the president and his campaign focused on two different candidates in the weeks before Democrats cast their first votes in Iowa — one, a populist running with a grass-roots movement behind him, and the other, a liberal billionaire who could refashion the general election because of his bottomless bank account.

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.