According to a video tweeted by the Guam News on Friday, Calvo while talking to Trump via an office phone switched the call to speaker phone to enable viewers hear the conversation clearly, Xinhua news agency reported.
Guam is telling its residents what to do if North Korea launches a missile attack at the USA territory, but is also calling for calm.
"So, with all the criticism going on over there for a guy who's doing what's positive, we need a president like you".
Before speaking to Mr Calvo, Mr Trump told reporters in New Jersey that he believed Guam would be "very safe" and threatened North Korea with "big, big trouble" if it launched an attack.
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"You've become extremely famous", he told Mr Calvo of the global news coverage following North Korea's threat. All over the world they're talking about Guam and they're talking about you.
"When (Russian President Vladimir) Putin says that, you know, he didn't interfere in our election - he was lying. I'll talk to you anytime you need, and again, it looks like just a handsome place".
Calvo agrees. "It's paradise, " he says.
Hawaii has also sought to prepare residents for the threat of a nuclear missile attack from North Korea, while stressing that the risk of a strike is low.
"If an attack warning is issued, take cover as quickly as you can, under concrete structure or below ground if possible", the sheet adds.
Dow streak of records ends amid US-North Korea tensions
The dollar index.DXY, which tracks the greenback against six rival currencies , was down 0.17 percent to 93.391. Mr Trump's words had come overnight Tuesday, UK time , in an escalation of tensions between the two countries.
Referring to an "imminent missile threat", it includes advice like: "Do not look at the flash or fireball - it can blind you".
"Don't worry about a thing and they should have had me eight years ago, or at least somebody with my thought process, because that was the time", Trump said.
Gen. Restituto Padilla Jr. said the possibility that the Philippines could be hit if North Korea proceeded with its threat was "remote".
Padilla made the statement after the North Korean Strategic Force of the Korean People's Army said on Thursday it was "seriously examining a plan" to fire a missile near Guam.
US, North Korea have engaged in backchannel diplomacy
Trump again on Friday warned North Korea against following through on a threat made Wednesday to fire a weapon against the U.S. "He has genuine concern about the direction of events in North Korea".