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 criticising 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.

But at the same time, Republicans have backed Trump's immigration plans since his return to the White House, without 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.

In the face of public outcry over the killings, Senator Rand Paul of Kentucky, the chairman of the Senate Committee on Homeland Security & Governmental Affairs, called on Monday for the leaders of ICE and two other federal agencies to testify before his panel next month. The request came alongside calls from other Senate Republicans for more oversight of ICE after Pretti's shooting.

Senator Lisa Murkowski of Alaska wrote on social media, ICE agents do not have carte blanche in carrying out their duties. Her condemnation does not come as a major surprise, as she is one of a small handful of Republicans in Congress who've been willing to openly criticise Trump.

Moderate Republicans like Murkowski and Trump's allies are beginning to voice concern about the immigration operation in Minnesota, indicating a significant shift in the party's stance.

The growing criticisms come as Trump shifts his own tone to ease tensions in Minneapolis, having previously described one shooting victim as a gunman. Recently, he has aimed to deescalate the situation, sending White House border tsar Tom Homan to oversee the operation there.

Despite the cautious rebukes from prominent Republicans, the party's future approach to immigration remains uncertain as they balance the need for accountability with their base's support for Trump's policies.