The retiring Tennessee lawmaker also said White House staff struggle to "contain" Mr Trump .
Mr Corker 's comments are a remarkable attack on a sitting president from a member of his own party.
Donald Trump has gone after members of his own party before. Most, however, have chosen to grit teeth and suffer in silence.
Not Bob Corker . Maybe there's something liberating about knowing you don't have to face an angry pro-Trump electorate next year. Maybe, as head of the foreign relations committee, the Tennessee senator feels compelled to condemn the havoc he believes Mr Trump is causing to US foreign policy.
The problem for the president is that, although Mr Corker is retiring, he's not leaving the Senate until January 2019.
Until then, he's chair of a powerful committee and a key vote in a chamber where only few party defections will sink any Republican-backed legislation.
The president may be settling scores with the Tennessean, but it could come at a high price.
"I don't think he appreciates that when the president of the United States speaks and says the things that he does, the impact that it has around the world, especially in the region that he's addressing," he said.
"I don't know why the president tweets out things that are not true," he added.
"You know he does it, everyone knows he does it, but he does."
"I know for a fact that every single day at the White House, it's a situation of trying to contain him," Mr Corker told the newspaper.
Mr Corker , who denied the president's account, responded: "It's a shame the White House has become an adult day care center.
"Someone obviously missed their shift this morning."
Mr Trump is this week expected to de-certify the Obama administration's 2015 agreement to curb Iran's development of nuclear weapons, a deal he has repeatedly condemned.
Last month Mr Corker - who has said the Iran deal should not be torn up - announced he would not seek re-election at next year's mid-term elections.
Mr Corker and Mr Trump previously clashed in August when the senator criticised the president's response to clashes that month between white supremacists and anti-fascist demonstrators in Charlottesville, Virginia.
Mr Corker said America's top diplomat was "in an incredibly frustrating place", where he "ends up not being supported in the way that I hope a secretary of state would be supported".
On Wednesday Mr Tillerson denied rumours that he was about to resign, amid reports he had referred to the president as a moron.