Speaking on "State of the Union," Kelly said President Donald Trump has told him to focus on two specific issues related to illegal immigration, and neither include deporting so-called dreamers, or undocumented immigrants who were brought into the United States as children and grew up in the country.
"The President told me to do two things," Kelly said. "He told me to secure the Southwest border -- all of our borders, and, of course, focusing now on the Southwest border -- and to take the worst of those that are in our country illegally, take them -- look for them and deport them. So that's what I'm doing."
Earlier this week Trump remarked that dreamers should "rest easy," the latest statement of his to indicate his administration won't take a hardline approach to dealing with the group.
Speaking about his immigration policy with The Associated Press
, Trump said his administration is "not after the dreamers, we are after the criminals."
Kelly told CNN that his "plea" to Congress is to pass legislation to deal with the issue.
"My message -- really, my plea -- and I've said this a number of times both before the camera as well as in hearings and virtually every single office call I have with members of Congress: Solve the problem legislatively," Kelly said.
"In the meantime, these people are caught between the law," he said. "We're not targeting them. The President obviously is sympathetic. But I just wish these kind of issues were dealt with legally by the United States Congress.
President Barack Obama placed protections on the group in 2012 through a policy known as Deferred Action for Childhood Arrivals (DACA), which shielded dreamers from deportation.
Although Trump has issued executive orders to strengthen the US-Mexico border and deport more illegal immigrants, his position on DACA has been unclear.
In February, Trump told reporters
, "DACA is a very, very difficult subject for me," and he has since mentioned that he plans to deal with the issue with "great heart."
Funding for a border wall
Kelly also said in the interview that he believes Trump "will be insistent" that lawmakers include money for a US-Mexico border wall in a spending bill that they need to pass by Friday to avoid a government shutdown.
"I think it goes without saying that the President has been pretty straightforward about his desire and the need for a border wall," he told Bash. "So I would suspect he'll do the right thing for sure."
Kelly said he thought Trump "will be insistent on the funding" for the wall, a lingering question ahead of the spending battle lawmakers face as they work this week to keep the government's lights on past Friday.
But a senior administration official told CNN on Saturday that the White House would not let the government shut down over the issue.
"The White House is not going to allow the government to shut down," the official said. "We've been clear about what our priorities are. Leadership in both chambers understands that."
Kelly also commented on the timing of the wall's construction Friday.
"We hope to begin construction by the end of the summer," he told CNN's Kate Bolduan on "At This Hour." "Clearly, we're not going to build a wall in an afternoon."
During a tour of the border last week, Kelly and Attorney General Jeff Sessions met Thursday with law enforcement personnel from several federal agencies at a federal building in El Paso, Texas, that houses immigration courts and is blocks away from a detention facility.
Kelly said he spoke with local and state lawmakers willing to share their views on the best way to build the wall.
Putting up a wall along the US-Mexico border was one of Trump's key campaign promises, and he issued an executive order in January directing that construction begin.
The Trump administration has already asked Congress for money to start building the wall, but Senate Democratic Leader Chuck Schumer and other top Democrats oppose the wall and say adding any money to the spending bill for it is a nonstarter.
Office of Management and Budget Director Mick Mulvaney has said he believes the administration and Republican leaders in Congress will be able to avert a government shutdown.
"We have our list of priorities," Mulvaney said Thursday at an event hosted by the Institute of International Finance, according to The Washington Post. "We want more money for defense. We want to build a border wall. We want more money for immigration enforcement, law enforcement."
Mulvaney told reporters Friday, "I do not think the government is going to shut down."