Republican lawmakers are starting to break with the Trump administration over its immigration crackdown in Minneapolis, as the party scrambles to respond to growing public anger after two US citizens were killed by federal agents.

But even as some Republicans in Washington speak out against the fatal shootings, they've avoided directly criticizing President Donald Trump or his broader immigration agenda.

The emerging messaging on Minneapolis points to the party's main dilemma heading into the midterm elections: whether and how to distance itself from the biggest controversies of Trump's second term, while running on his overall record on immigration and other issues - a record that's popular on the right, and that most Republicans helped push through Congress.

So far, Republicans have focused criticism on US Immigration and Customs Enforcement, or ICE - the agency at the forefront of the Minneapolis operation - and other agencies within the Department of Homeland Security taking part.

At the same time, Republicans have backed Trump's immigration plans since his return to the White House, but now avoid highlighting their support.

Last year, the Republican-controlled Congress approved roughly $45 billion for border security and additional funding for interior enforcement to help the administration carry out Trump's plan to deport millions of undocumented migrants from the country. That support continued as opinion polls have shown a majority of Americans believe ICE has gone too far in its tactics under Trump.

In the face of public outcry over the killings, Senator Rand Paul called on ICE and two other federal agencies to testify before his panel next month, coinciding with calls for more oversight after the shootings.

Moderate Republicans aren't the only ones in the party criticizing the operation; some of Trump's allies in Congress have also begun to speak out, framing their comments to avoid explicit references to the president.

As responses to the situation continue to evolve, the party's strategy on immigration and its future messaging as the midterms approach remains uncertain.