Bolton was siding with the president, who has angrily blamed Nielsen for failing to staunch the flow of migrants across the border, while Kelly, who is protective of his protégé, defended her, the people said.
Speculation has also reportedly arisen within the White House that Kelly or Bolton may step down soon.
US President Donald Trump is aware of the argument between two of his most senior staff, sources say.
The argument revolved around difference of opinion over the White House's response to the drastic influx of migrants crossing the southern border.
This comes after reports claimed the two administration officials had a shouting match outside of the Oval Office. Kelly then stormed away, so upset that he uttered some profanities. Both believed the goal was to protect the border.
G League to offer $125K contracts to elite prospects
That monopoly on young talents may be over with the announcement from the National Basketball Association G League today. Turner made it clear that the G-League would not pursue any players who have already committed to play for a college.
Machado fined by Major League Baseball for kicking Aguilar
The shortstop later scored the winning run, sliding home to give Los Angeles a 2-1 win that evened the NLCS at two games each. After the game, Machado intimated that he and Aguilar are longtime friends and called him "a great guy".
Saudi Arabia's full statement on the death of journalist Jamal Khashoggi
President Donald Trump , speaking at a defense roundtable in Arizona on Friday night, called the Saudi report a "good first step". Saudi intelligence officials have reportedly been dismissed in the aftermath of his disappearance on October 2 .
Trump threatened yesterday to deploy the military and close the southern U.S. border as Hondurans and Salvadorans joined thousands of migrants in Guatemala hoping to travel north.
The clash is an indication that tension is flaring in the White House 19 days before midterm elections in which Republican control of Congress is at stake. CNN reported on fears in the White House that someone might resign over the argument.
The fight allegedly took place earlier this week, as reports, citing unpublished statistics by the Department of Homeland Security, said that 16,658 family members were arrested trying to illegally cross into the United States in September, the highest one-month total on record and representing an 80-percent jump from July.
Sanders stated, "While we are passionate about solving the issue of illegal immigration, we are not angry at one another".