Love later issued an updated statement, removing references to an "overly broad executive order" and mention of worry that it may "undermine" the immigration system.
Trump's order, temporarily barring travelers from six Muslim-majority countries and indefinitely halting those from Syria, has brought swift rebukes and legal challenges since he signed it Friday. More than 100 passengers flying into the United States from those countries last weekend were detained at airports while federal judges stayed the order's implementation pending further court action.
The White House on Monday defended the order, which follows Trump's campaign promise to ban Muslims from entering the United States, as an extension of travel restrictions called for by then-President Barack Obama, though through a spokesman the former president said he "fundamentally disagrees with the notion of discriminating against individuals because of their faith or religion."
In Love's updated statement, she said America "must secure our nation's borders and ensure Americans are protected while continuing to welcome individuals and families pursuing the American dream."
She added that Trump is "rightly" focused on national security but Congress must play a significant role in crafting immigration policy. "Our message to our friends and allies around the world and here at home regarding immigration must create clarity, not confusion," she said.
Rep. Jason Chaffetz, R-Utah, told reporters at an event with wealthy conservatives hosted by the Koch brothers that he opposes basing immigration decisions on religion but he generally supports Trump's efforts.
"I understand the need and support the idea that vetting from certain countries is needed," Chaffetz said this weekend, according to The Washington Post. "A legal permanent resident I think is a different category. I don't understand what they're trying to do on those categories. People that have a green card supposedly have been vetted. So there needs to be some further clarification."
Chaffetz also praised Trump for taking action as he vowed to do in his White House run.
"He's off to a roaring start," Chaffetz said, according to The Post. "I think it's surprising a lot of people that he is actually doing what he said he was going to do, but there are those of us that actually support that."
Sen. Mike Lee, at the same event in California, didn't respond to reporters' questions about the refugee moratorium, The Post said.
Instead, Lee's office issued a statement saying that the GOP Utah senator is pleased the White House is focused on protecting the American people, though he wants to learn more about the details of how it will be implemented.
"I do have some technical questions about President Trump's executive order titled 'Protecting the Nation From Foreign Terrorist Entry Into the United States,' " Lee said. "My staff and I will continue to reach out to the White House for clarification on these issues."
The office of Rep. Rob Bishop, R-Utah, did not respond to questions about the president's order.
Rep. Chris Stewart, R-Utah, said last week that he supports efforts to halt some refugee intake because, he argued, there is no way to vet some immigrants, including those from Syria.
Monday, Stewart doubled down on support for a moratorium, saying that he's traveled to war-torn areas of the world and, as a member of the House Intelligence Committee, has attended several classified briefings on terrorism.