NBC News adds questions about Kelly's "future in the White House recently became more serious after his repeated clashes with national security adviser John Bolton and his deputy, Mira Ricardel". One person familiar with the matter said Ricardel quarreled with the first lady's staff over seating on the plane and use of National Security Council resources.
President Donald Trump has made a decision to replace Department of Homeland Security Secretary Kirstjen Nielsen in the coming weeks after becoming increasingly frustrated with her performance on immigration enforcement.
On Tuesday, first lady Melania Trump publicly called for the ouster of a senior member of her husband's administration.
Ricardel has denied trying to undermine Mattis.
Ms Ricardel was also believed to have had a dislike of US Defence Secretary Jim Mattis, current and former officials said.
Melania allegedly complained to the President, 72, saying she was unhappy with how she was treated by Ms Ricardel, according to two sources speaking to Reuters.
Trump has told advisers that he wants to replace Nielsen, according to the Post, and that she may leave office in the coming weeks, or perhaps much sooner.
The constant fighting almost led to Nielsen's resignation, but she was talked into staying be her mentor John Kelly.
Kelly may be replaced with Nick Ayers, who is now chief of staff to Vice President Mike Pence, the report added.
A source claimed Nielsen - who used to work on disaster management response for the White House under Bush - is referred to as a "Bushie" by Trump.
Still, Trump is said to have blamed her for a rebound in the number of people arrested at the Mexican border this year.