Democrats, meanwhile, were sharply critical of the parade proposal, arguing it was a waste of money and a vanity exercise for Trump.
A White House official said there is "brainstorming" about the parade but nothing is settled.
The parade, one of a number of parades and celebratory events held in the wake of the war, was reportedly the largest victory celebration since the end of the Second World War.
After talking about the need for increased military funding, Mattis would not say how much Trump's military parade would cost.
Press Secretary Sarah Sanders said Trump's intention was to have "a celebration" of the military. "He has asked the Department of Defense to explore a celebration at which all Americans can show their appreciation", Sanders said. The parade also commemorated the 100th anniversary of the United States' entry into World War I in 1917.
When Mr Trump met with the French President Emmanuel Macron in September, the commander-in-chief gushed about seeing France's military might on display in the streets when he watched the Bastille Day parade with his French counterpart in July.
"So we're actually thinking about Fourth of July, Pennsylvania Avenue, having a really great parade to show our military strength", Trump said months after his Paris experience.
Republican Sen. Thom Tillis of North Carolina, who sits on the Senate Armed Services and Veterans' Affairs committees, said such a parade would send a "great statement" of thanks to the military and could inspire those who want to join. Is a national military parade a good idea?
Retired Maj. Gen. Paul Eaton, an adviser to VoteVets an organization focused on electing veterans to public office, criticized the president's idea, saying Trump "has has continually shown himself to have authoritarian tendencies and this is just another worrisome example".
Sen. John Kennedy, a Louisiana Republican, said the world already knows that the USA has the strongest military in the world, and that there's no need to broadcast it. The parade was then held on the Constitution Avenue.
Trump was French President Emmanuel Macron's guest on Bastille Day previous year, and later called the French military parade he witnessed "one of the greatest parades" he had ever seen. John Kennedy, R-Louisiana.
"It's a morale booster for our military organizations", Harris said.