Among the issues Trump is willing to address is legal status for those who are in the country illegally but have not committed crimes.
Trump would not necessarily support a pathway to citizenship - possibly except for "Dreamers," who were brought into the country illegally as children - according to a report by CNN's Wolf Blitzer and Jake Tapper, who attended the luncheon.
Sarah Huckabee Sanders, the White House principal deputy press secretary, said she could not confirm Trump's comments in his private luncheon.
"The President has been very clear in his process that the immigration system is broken and needs massive reform and he's made clear that he's open to having conversations about that moving forward," Sanders said in a briefing with reporters. "Right now his primary focus, as he has made [clear] over and over again, is border control and security at the border."
Trump has vowed to build a wall along the US-Mexico border and increase funding for federal law enforcement efforts in border areas. He also has instructed the Department of Homeland Security to round up and deport illegal immigrants who have committed serious crimes or caused violence.
"We're getting gang members out, we're getting drug lords out, we're getting really bad dudes out of this country - and at a rate that nobody's ever seen before," Trump said last week. The President likened the raids to "a military operation," though Homeland Security Secretary John Kelly made clear the military was not involved in conducting them.
It is unclear whether Trump will follow through on his stated openness for an immigration compromise. The President has in the past made comments, in private or in media interviews, that are not borne out by his Administration's policies. For example, he has yet to follow through on his pledge to investigate alleged voter fraud in the 2016 election.
In recent weeks, Trump has expressed openness to revisiting past immigration reform efforts, including the failed 2013 "Gang of Eight" bill, which drew opposition from Republicans.
At a meeting with moderate Democratic senators, Trump told them he thought the Gang of Eight bill was something he was interested in revisiting, according to the senators.
"He's open to reviewing the piece of legislation," Senator Joe Manchin said. "He says, 'Well, you've got to start working on it again,' and I said, 'Absolutely we will.' And that was encouraging."
The White House later denied that Trump was open to the legislation and said that he considered the bill to be "amnesty".