"Campaigning is very different from governing," said Mr Kelly on Wednesday.
Hours later, the president tweeted to say Mexico would still "directly or indirectly" pay for the wall.
Mr Kelly had also reportedly described the president as "uninformed" when he made campaign promises about a wall. The comments were overheard during an immigration meeting on Wednesday, according to US media.
Why does the wall matter now?
Immigration has become a key issue between lawmakers, and the White House is currently risking a federal government shutdown from midnight on Friday.
Congress faces the deadline to pass a stopgap measure that would fund federal agencies until next month.
Democrats want the bill to include protections for immigrants who entered the US illegally as children, known as "Dreamers".
Republican President Trump has been fighting to scrap the Deferred Action for Childhood Arrivals (Daca) programme.
However, he had signalled he was ready to make a deal to help the Dreamers, in return for funding for border-security plans, which include building the wall along the Mexican border.
Bipartisan immigration talks had sounded promising until it was reported last week that Mr Trump had dubbed certain nations "shitholes" during legislative negotiations.
Have the wall plans changed?
Mr Kelly said the administration was now looking at creating a wall across 800 miles (1,300km) of the 3,100km stretch. He said this would include improving existing fences.
He said the estimated cost was $20bn (£14bn; 16bn euros); Mr Trump had originally put the figure at $10-$12bn.
During his election campaign, Mr Trump had also insisted Mexico would pay for it in its entirety.
Mr Kelly said they were now looking at alternative ways to raise funds, including via possible visa fees or renegotiating the Nafta trade deal.
"Campaigning is very different from governing," he said in the Wednesday night interview.
Earlier, other US media reported that Mr Kelly had privately told a group of Democrats from the Congressional Hispanic Caucus that President Trump had not been fully informed when he came up with his wall plans.
Mr Kelly was quoted as saying that he had tried to persuade the president to change his position on the issue.
Neither Republicans nor Democrats want to be blamed for a federal shutdown with crucial mid-term elections looming in November.
Republicans have the advantage, by controlling both chambers of Congress, but they are seen as divided on key issues. Some oppose the current bill as they want more money for defence, others oppose the idea of another stop-gap, rather than finalising a longer-term agreement.
BBC News North America reporter Anthony Zurcher says if they stick together, the Republicans in the House of Representatives can pass some sort of short-term solution without any Democratic support. However, Senate Democrats will then have to decide whether they have the numbers to block the bill and force a shutdown.
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