Sources with direct knowledge tell TMZ ... Kelly arrived last Friday just before 8 AM -- already in full glam -- and she and her team began complaining about her dressing room from the jump. We're told "Today" staffers explained their dressing rooms were notoriously "dumpy, glorified closets."

The sources go on to say Kelly and her team were "quite upset," and asked who had the dressing room upstairs. They were told that room was taken by J Lo, but they were then offered a few options. We're told Kelly and her team liked none of them, and 7 minutes before air, they said, "We're gonna leave," and they booked it.

The sources say there's no truth to the notion Kelly's breaking point was Savannah Guthrie repeatedly asked KR about Beyoncé during her sit-down in the second hour of the show.

These reports claim Kelly has a stipulation for interviews that Beyoncé and Destiny's Child questions were off limits. Whether that's true or not ... we've been told squarely that's not why Kelly left.

That jibes with what the main characters have said over the last couple of days ... including Hoda herself, who explicitly mentioned dressing rooms Monday while extending an olive branch to Kelly.

Also ... Sherri Shepherd leaned into the dressing room dust-up while welcoming Kelly on her own show ... and they played it up further with BTS footage, showing the dressing room Kelly got -- and telegraphing it was way better than what "Today" offered.

Now, on the issue of Kelly being peeved about questions regarding Beyoncé/DC -- there may actually be some truth to that ... as she did come across as a bit annoyed with Savannah, and she proved to be annoyed in other interviews this week too.

With that said ... that's not why she left "Today" on Friday. It was, indeed, dressing room problems. Now, that's not to say Kelly isn't welcomed back to "Today" -- 'cause she is.