Calvo, a fellow Republican, responds, "Mr. President, as the governor of Guam representing the people of Guam and as an American citizen, I have never felt more safe or so confident than with you at the helm".
North Korea recently threatened the Pacific island with a missile attack.
Before the call ended, Calvo invited Trump to Guam, and Trump told him to come see him in the White House the next time he's in Washington.
With Guam's safety in the balance, Trump assured the territory's Governor Eddie Calvo: "We are with you 1000 percent, you are safe".
"I'll talk to you any time you need", Trump said.
I take issue with your editorial that includes a description of "residents of Guam glancing nervously skyward" ("Alternative fact of the Week: Trump's nuclear arsenal", August 10).
"You know, they're showing so much, it's such a big story in the news and it just looks like a handsome place", he says.
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Despite Mr Trump's reassurances, Guam's security agency has issued a fact sheet with advice on what residents should do in the event of a nuclear attack.
This is not to say that North Korea will not respond to President Trump's remarks with plenty of blood-curdling threats, but it is important to note that the threat universally cited by the mainstream media as Kim Jong-un's terrifying push back against Trump's bluster was, in fact, a threat issued for entirely different reasons - both of them involving American military forces located on USA soil.
Mr Trump has said he would not allow Pyongyang to develop a nuclear weapon capable of hitting the United States. "But you're going to be taken care of".
"Good morning, good morning", the president is heard saying to Calvo.
The president called the U.S. military "incredible, "rock solid" and "the best in the world by a factor of five".
Key Pyongyang ally Beijing, meanwhile, has pleaded with Trump to tone down his rhetoric to prevent tensions from boiling over.
Guam residents found themselves practicing air-raid drills in early 2013 after a previous round of threats, in which North Korea specifically named Andersen Air Force Base as a target.
The North's official KCNA news service, for its part, accused Trump in an editorial of "driving the situation on the Korean peninsula to the brink of a nuclear war", calling the United States "the heinous nuclear war fanatic".