He played down criticism of his administration's response to the hurricane by stating that "the military and first responders. have done an incredible job".
Hurricane Maria struck Puerto Rico on September 20 as a Category 4 storm with winds of 155 miles per hour, the strongest to hit the island in a century.
Cruz appeared to respond Saturday, though she did not mention Trump by name.
"The Mayor of San Juan, who was very complimentary only a few days ago, has now been told by the Democrats that you must be nasty to Trump".
Mr Trump came under withering fire earlier this week for what some critics said was his slow response to an escalating humanitarian crisis.
Tom Bossert, Trump's homeland security adviser, who joined Duke in addressing reporters, said some of the media reports about the island's condition had been based on outdated images and data.
"I am begging, begging anyone that can hear us, to save us from dying", Cruz said at a news conference Friday.
"We are dying here". Trump will visit the island on Tuesday. The idea here is clear; America citizens living in Puerto Rico should be on their knees thanking Trump for doing anything at all.
US Vice President Mike Pence also defended his boss's performance by taking to Twitter. He said she had "poor leadership ability". "We want the people to be safe and sound, and we will be there every day until that happens".
"It is unfortunate that the San Juan mayor wants to sort of go against the grain", Mulvaney said on CNN's "State of the Union", adding, "We'd love to have her on the team as we all pull in the same direction".
"The people who are the best in this are the USA military".
Sanders characterized the president as "speaking from his fancy golf club, playing golf with his billionaire friends, attacking the mayor of San Juan".
Trump has been under fire over the federal government's response to the situation in Puerto Rico in the wake of Maria, the tenth-most intense Atlantic hurricane on record, the worst to strike Dominica in its recorded history, and the strongest in Puerto Rico since 1928. That rule was preventing foreign ships carrying aid and fuel from docking in Puerto Rico.