A report by Politico that President Trump set a July 4 deadline for Chief of Staff Reince Priebus to get the White House in order was confirmed by CBS News Sunday, according to a source familiar with the situation. However, CBS News' Major Garrett's reporting indicates that this is not so much a date circled in red as it is a time to evaluate Priebus' progress with the Trump legislative agenda.

The GOP-controlled Senate is under pressure to vote on its version of a bill to repeal and replace Obamacare. Republicans have been holding secret talks to write their measure, even though they lambasted Democrats for drafting the Affordable Care Act (ACA) in a secretive fashion. They aim to vote on their bill before the July 4 recess.

At this moment, Mr. Trump is reluctant to shake up his staff because it would only add to the perception of chaos and instability in the White House, Garrett reports. It had been discussed that former deputy campaign manager David Bossie could be named deputy White House chief of staff, a position still vacant since Priebus ally Katie Walsh was pushed out in March. At that time, it was thought that her dismissal was a bad sign for Priebus' job security, but it's been more than two months, and he's still in place.

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According to Politico, which on Sunday night first reported a July 4 deadline for Priebus, the president was angry over the White House dysfunction and chastised Priebus in front of former campaign manager Corey Lewandowski and deputy campaign manager David Bossie. Garrett spoke with two separate sources present at the Memorial Day meeting, and they disputed that the president mentioned the July date in relation to Priebus. The president did, however, discuss Priebus' and others' performance and said that the chief of staff was under scrutiny.

Mr. Trump had considered bringing in Lewandowski and Bossie to spearhead White House messaging, including issues like the Russia investigation, but Mr. Trump decided against the entire "war room" set up with Bossie and Lewandowski because he concluded that if either or both of them join the Trump team in the White House, they should have bigger roles than dealing with Russia-related stories. Instead, the war room was instead given to two people removed from the Trump White House: Walsh and former communications director Mike Dubke.

According to Politico, the White House denied the existence of the July 4 deadline -- press secretary Sean Spicer said, "Whoever is saying that is either a liar or out of the loop."

For the second straight week, the Trump White House is promoting a theme -- this week is "Workforce Development" -- while last week, the administration was busy promoting "Infrastructure Week." This reflects a new internal message and policy discipline on the part of the communications team, Garrett suggested, giving some of the credit for this to Priebus.

CBS News Political Director Steve Chaggaris contributed to this report.