"Anything that we can do to help the people that are sacrificing so much for us is just so critically important", Howard Somers said.
But it is up to Mr Trump to sign off on all information to be released, because of medical privacy issues. He will not undergo a psychiatric exam, the White House said.
Past presidential physicals mainly paint a portrait of vigor - punctuated by details like workout routines and sports injuries - but Trump isn't known to exercise beyond short walks.
His remarks come in the wake of a book "Fire and Fury: Inside the Trump White House" which is highly critical of him and the First Family.
On Tuesday night, CNN asked Cohen if he had told Trump or White House officials about the suit ahead of time, and Cohen declined to comment.
United States media are playing down expectations the medical will silence the debate over the president's mental health.
"A lot of focus on the veterans", said Trump. In George W. Bush's first exam as president, his doctor said he enjoyed the occasional cigar, drank diet soda and ran 12 miles a week.
GETTY TURNING POINT Donald Trump's first physical exam could be a major event
The APA said it has confidence that Trump's physician will conduct an age-appropriate medical and mental health evaluation.
The book quotes former advisers and others who paint an unflattering portrait of Trump and his fitness to be president.
Jackson, a rear admiral, has served as a White House physician for the past three administrations, according to the U.S. Navy.
Press Secretary Sarah Huckabee Sanders did say that results of Trump's January 12 physical will be released after its completion, but she was unclear what that would entail.
Trump's personal doctor, Harold Bornstein, released a letter during the presidential campaign claiming the real estate mogul would be the "healthiest individual ever elected" and that his laboratory tests were "astonishingly excellent".
Trump takes Crestor for his cholesterol, a low-dose aspirin for heart attack prevention, Propecia to treat male-pattern baldness and antibiotics for rosacea.
By the amendment's terms, if the vice president and a majority of the Cabinet determine the president is "unable to discharge the powers and duties of his office", the vice president immediately becomes acting president.
Meanwhile, the president also addressed his latest call with South Korean President Moon Jae-In, who Mr. Trump says, was thankful for the role the US played in the talks between North Korea and South Korea this week over the North's participation in the Winter Olympics.