In an interview with CNN's Dana Bash on "The State of the Union," Kelly said he's confident Trump will "do the right thing."
"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 other senior administration officials stopped short of saying the President would press hard for the funding, sending mixed signals as lawmakers approach what is expected to be a busy week.
In an interview on "Fox News Sunday," White House budget director Mick Mulvaney did not guarantee a shutdown threat, saying funding for the wall was one of several subjects being negotiated.
When asked by anchor Chris Wallace if Trump would sign a bill that doesn't include funding for the wall, Mulvaney said: "We don't know yet. We are asking for our priorities, and importantly, we are offering to give Democrats some of their priorities as well."
But, he added, "Shutdown is not a desired end. It's not a tool."
White House chief of staff Reince Priebus said on NBC's "Meet the Press" that he was "pretty confident we're going to get something that is satisfactory to the president in regard to border security within current negotiations."
When asked by moderator Chuck Todd whether the White House would be satisfied if money isn't specifically appropriated for the wall itself, Priebus added: "I think that as long as the President's priorities are adequately reflected in the CR (continuing resolution), and it allows us to get moving with an increase in military spending and a rebuilding of our military as he promised in one of your bullet points, and there's enough as far as flexibility for the border wall and border security, I think we'll be okay with that. But we're still negotiating this weekend with appropriators in the House and the Senate."
However, 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."
Sen. Dick Durbin, the No. 2 Democrat in the Senate, said on CNN's "State of the Union" that a shutdown over the wall funding, which Trump had promised Mexico would pay for, would be a poor political stunt.
"That would be the height of irresponsibility," Durbin said. "He would not want that to define his first 100 days."
Still, Kelly gave a quick timeline for the start 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, which includes 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 Trump 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."