His comments on Twitter came a day after Kelly told Democratic lawmakers on Capitol Hill that some of Trump's hard-line immigration stances were "uninformed", that Mexico would never pay for the wall and that the wall would not stretch across the entire southern border.
Mr Kelly said on Fox he told the caucus that "they all say things during the course of campaigns that may or may not be fully informed".
Talks have fallen apart following Trump's reported derogatory comments at a White House meeting last week targeting immigrants from Haiti and African nations as being from "sh-hole" countries, remarks that drew worldwide condemnation.
Both Trump and Kelly may have been able to dismiss those recollections as fake news, had Kelly not made the fatal mistake of repeating the claims on Fox News-an outlet all but guaranteed to draw the attention of its Oval Office assignment editor.
A border wall between the United States and Mexico is needed, Trump said, "for the safety and security of our country".
"His only frustration is with the media and that's what that tweet was about".
Trump and Kelly "have a great relationship and that continues", Shah added.
Kelly has repeatedly said that he doesn't pay attention to the president's tweets.
Trump detailed that the payment by Mexico would be through a long-term reimbursement and described as ridiculous the commercial superavit of that country of 71 billion dollars with Washington.
"I'm looking for something that President Trump supports, and he has not yet indicated what measure he is willing to sign", Senate Majority Leader Mitch McConnell (R-Ky.) said Wednesday. "He's a really special guy". Well, I don't think he said it the way you would like him to say it. "The $20 billion dollar Wall is "peanuts" compared to what Mexico makes from the U.S. NAFTA is a bad joke!" referring to the North American Free Trade Agreement that former US President Bill Clinton, negotiated during the 1990s. "But what Kelly is talking about is him listening to experts and folks on the front lines, which is something he said he would do throughout the campaign", Shah said.
The border wall stands at the centre of a congressional struggle to overhaul immigration before the March expiration of a programme protecting so-called "Dreamers" - immigrants who were brought to the country illegally as children.
"T$3 here are places where hydrographically, geographically, a wall would not be realistic".
Specifically, Rep. Luis V. Gutiérrez recalled Kelly as saying that "there were statements made about the wall [during the campaign] that were not informed statements".
"In one way or another, it's possible that we could get the revenue from Mexico, but not directly from their government", he said.