Washington (CNN) Secretary of State Rex Tillerson acknowledged Thursday that the process of staffing senior positions in his agency has progressed slower than he'd like, as the State Department and White House trade blame over the issue.

Asked whether he was satisfied with the pace of staffing for these jobs, which include nearly all undersecretary and assistant secretary positions as well as dozens of ambassadorships, Tillerson responded simply: "No, I'd like to go faster."

The comment comes after a testy exchange at the White House on Friday between Tillerson and Johnny DeStefano, who leads the presidential personnel office, which was first reported by Politico . Two officials familiar with the meeting, which took place in White House Chief of Staff Reince Priebus' office, confirmed to CNN that Tillerson let his frustrations over the process be known.

They say Tillerson was unloading his frustration over the pace of appointments, and what he sees as White House interference, on DeStefano but also on Priebus and Jared Kushner, the President's senior adviser and son-in-law, who was also present.

While it is not unusual for these kinds of clashes to occur between Cabinet officials and the White House personnel, one official familiar with the meeting described this particular conversation as "intense" and "uncomfortable."

The White House did not respond to a request for comment about the meeting.

Asked about it on Thursday, RC Hammond, a senior communications adviser to Tillerson, told CNN, "colleagues are capable of frank exchanges."

"The purpose of the meeting was to help expedite the process of appointments and help everyone make the best decisions they can," he added. "Each side took the time to make their case and did a good job of listening to each other's concerns and better understand how each can do their jobs and that will result in the process picking up."

Officials tell CNN that, for weeks, Tillerson and his top advisers have been frustrated with DeStefano over the pace of appointments and attempts by the White House to bring in their own candidates at the expense of those chosen by Tillerson's team.

"Evaluating nominees did get off to a slow start, and everyone acknowledged that," Hammond said. "White House personnel would like it to go faster and Tillerson is not satisfied and has said he would like to it to go faster."

"Since Deputy Secretary (John) Sullivan has come aboard to help evaluate candidates there has been an uptick in nominations and nominations are moving along at a faster pace," he noted, referring to Tillerson's No. 2 at the department, who was confirmed by the Senate last month.

One key cause of the hiring delay, two officials tell CNN, is a loyalty test -- a message permeating from the White House that people hired within the administration must be sufficiently loyal to Trump, and not have said anything against him.

Earlier this year, the White House nixed Tillerson's top pick for the deputy Secretary of State post, Elliott Abrams, after President Donald Trump learned Abrams had been critical of him during the 2016 campaign.

But the rule doesn't appear to be absolute. Trump tapped former South Carolina Governor Nikki Haley -- who was sometimes critical of him during the campaign -- as ambassador to the United Nations.

And Hammond, the Tillerson communications adviser, tweeted criticisms of Trump during the primary season, including the hashtags "#vote #NotTrump," before ultimately joining the administration.

White House officials, for their part, have expressed frustration that Tillerson is taking so long to make appointments, and that his chief of staff, Margaret Peterlin, has jettisoned most of the names they've suggested, denying their candidates interviews.

DeStefano, these officials say, is among those who have voiced frustration to several colleagues.

A source familiar with the meeting said Tillerson voiced frustration about his lack of autonomy in choosing personnel and about the quality of candidates proposed by the White House, and questioned DeStefano's judgment.

Tillerson has faced resistance within the White House for relying on career civil servants.

For example, he wants to nominate long-time State Department official Susan Thornton for the position of Assistant Secretary for East Asian and Pacific affairs, a role she has held in an acting capacity for several months. But the White House has so far resisted that pick, in an effort to put a political appointee in that role.

"If the secretary is putting forward a nominee such as Susan, he is doing it because he believes she's the most qualified and best fit for the position," said Tillerson spokesman Hammond.

"In this particular case, he has the experience of working with her, since she has been on the job since February," Hammond noted. "That includes a trip to Beijing, a visit from the Chinese president to Mar-a-Lago, and the first of a series of four dialogues."

In a congressional hearing two weeks ago, Tillerson defended the department's efforts to fill vacant positions, and attributed delays to the "paperwork burden" that prospective nominees must shoulder as they go through the vetting process.

Previously, the State Department has also blamed the Senate confirmation process for holding up their nominees.