Greenland, with a population of around 57,000, is defended by Denmark, whose military is dwarfed by that of the U.S., which has a military base on the island. Denmark's prime minister has warned that a takeover would threaten NATO.
The U.K. agrees with Trump that Russia and China are increasingly becoming more competitive in the Arctic Circle, Alexander said.
"Whilst we haven't seen the appalling consequences in that part of the world that we've seen in Ukraine, it is really important that we do everything that we can with all of our NATO allies to ensure that we have an effective deterrent in that part of the globe against (Russian President Vladimir) Putin," Alexander told the BBC.
Britain's former ambassador to the U.S., Peter Mandelson, who was sacked last year because of his friendship with the disgraced financier Jeffrey Epstein, said he did not think Trump would take Greenland by force.
"He's not a fool," Mandelson said. "We are all going to have to wake up to the reality that the Arctic needs securing against China and Russia. And if you ask me who is going to lead in that effort to secure, we all know, don't we, that it's going to be the United States."