The group inducts "the best and brightest liberal arts and sciences undergraduates from across the nation" with presidents Bill Clinton, George H. W. Bush and Jimmy Carter being members of the exclusive club.
Dr Barbara Perry, the director of presidential studies at the University of Virginia told the BBC, Mr Trump's confidence hints at a high IQ.
"If he ever releases his IQ, I just have a feeling - especially since he is daring Tillerson to release his - that it's higher than people would presume," Dr Perry said.
"People who don't like him say 'oh he's such an idiot, oh he's so stupid'. But I bet you it's higher than we might realise."
Mr Trump's attendance at the University of Pennsylvania's Wharton School of Business also hints he could be quiet intelligent - something the President himself has reminded us of.
"I went to the Wharton School of Business," he said in a campaign speech in 2015. "I'm, like, a really smart person."
And while Mensa, the organisation for people with an IQ over 130, offered to administer the test themselves, Dr Frank Lawlis, the psychologist for American Mensa, said their testing isn't the be all and end all.
"If you take someone we consider to be a genius like Einstein, he would probably not do well on an IQ test, because he thinks outside the box," Dr Lawlis said.
"He could probably think of a dozen answers to one question."
But other experts don't particularly agree with Mr Trump's declaration.
Joel Schneider, a professor at Temple University, said the president appears to have a strong working memory - which is one of the basic components of a high IQ.
"His mind wanders as he talks a lot, but what's striking to me is that he usually finishes his sentences, no matter how long they are," Mr Schneider told The Guardian . "He doesn't lose track."
Other experts claim the way Mr Trump speaks hints at him having a small vocabulary, which is not a sign of intelligence.
When Mr Tillerson was asked point-blank in a press conference yesterday if he actually did call the president a "moron", the Secretary of State simply sidestepped the question.
"The places I come from, we don't deal with that kind of petty nonsense," he said.
"I'm just not going to be part of this effort to divide this administration."
To test your own knowledge, here's the world's shortest IQ test - which isn't as easy as it seems.
1. A bat and a ball cost $1.10 in total. The bat costs $1.00 more than the ball. How much does the ball cost?
2. If it takes five machines five minutes to make five widgets, how long would it take 100 machines to make 100 widgets?
3. In a lake, there is a patch of lily pads. Every day, the patch doubles in size.
If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake?
Dubbed the Cognitive Reflection Test, experts claim that getting all the answers right is a sign of genius.
ANSWERS
1. The ball would actually cost five cents.
If the bat costs $1 more than the bat, you need to think outside the box.
While it might be easy to assume that the ball is 10 cents, that would make the bat $1.10 as it costs $1 MORE than the ball.
2. It would take five minutes to make 100 widgets. Five machines can make five widgets in five minutes. This means that one machine will make a widget in five minutes, too.
3. It would take 47 days for the patch to cover half of the lake.
The patch is doubling in size each day it is going forward, so it would halve going backwards.
