The director of national intelligence serves as an overseer of the U.S. intelligence community and a close adviser to the president and National Security Council, producing each day's top-secret Presidential Daily Brief.

A source with direct knowledge told me that Trump has privately said he thinks the Office of the Director of National Intelligence represents an unnecessary bureaucratic layer and that he would like to get rid of it. He has been told that eliminating the ODNI is not politically possible, but still would like to "downsize" the office, the source said.

A government source who has discussed the matter with Trump characterized the president's thinking this way: "It's time for a change. Dan's a great guy but the president doesn't listen to him anymore."

A White House official responded: "We have no personnel announcements at this time."

The big picture: Coats has rankled Trump more than once with his public comments, according to sources with direct knowledge.

He angered Trump when he appeared to criticize the president's relationship with Russian President Vladimir Putin during an on-stage interview with NBC's Andrea Mitchell at last year's Aspen Security Forum.

He drew Trump's ire again in January when he told a Senate panel that North Korea was unlikely to give up its nuclear weapons, contradicting the president's cheerier assessments.

The other side: In a statement provided by the ODNI, Coats said, "I am focused on doing my job, and it is frustrating to repeatedly be asked to respond to anonymous sources and unsubstantiated, often false rumors that undercut the critical work of the Intelligence Community and its relationship with the President. I am proud to lead an IC singularly focused on the vital mission of providing timely and unbiased intelligence to President Trump, Vice President Pence and the national security team in support of our nation’s security."

Coats previously served for 16 years as a senator from Indiana — a tenure bookended by a stint as the U.S. ambassador to Germany during George W. Bush's administration.

He's also close with Vice President Mike Pence, a fellow Hoosier. (The above statement from Coats was originally provided by the ODNI in response to an NBC News report in March that Pence had talked Coats out of resigning.)

What's next: One potential replacement Trump has mentioned to multiple sources is Fred Fleitz, who formerly served as chief of staff to national security adviser John Bolton.

Fleitz was previously a CIA analyst and a staff member of the House Intelligence Committee. He is currently the president of the Center for Security Policy.

Trump has told people that he likes Fleitz and has "heard great things." Fleitz has publicly criticized Coats and even called for Trump to fire Coats on Lou Dobbs' Fox Business program after Coats' Senate testimony. Fleitz accused Coats of undermining and "second-guessing" the president.

Go deeper: Dan Coats says he wishes Trump took a different approach in Helsinki