Source: Matt Lauer 'Fighting to Save His Marriage'

Despite all the allegations of sexual misconduct against him, sources close to Matt Lauer say he's still hopeful that he and Annette Roque can salvage their marriage.

Lauer and Roque have been married for 19 years, though there have been rumors of divorce surround them since long before Lauer was fired from the Today show. Roque even filed for divorce in 2006, and some say that Lauer paid her $5 million to drop the case.

Still, an anonymous source told PEOPLE that Lauer "is fighting to save the marriage."

"He doesn't want a divorce," the source said. "Both of them, their first thought is their children." Lauer and Roque have three children together — 16-year-old Jack, 14-year-old Romy and 11-year-old Thijs. The source said that the couple has agreed their number one priority is not to disrupt the lives of their kids, especially while they're in school.

Still, Roque and Lauer were both seen out in public with their wedding rings off this week, which many took as a sign that they were through. However, a second person close to them told PEOPLE that nothing has been decided just yet. They said that Roque is "angry, so she took off her ring. She wanted him to take off his ring, too."

The source said that try as they might, Lauer and Roque can't hide those kinds of details from their kids.

"The kids are noticing this, but they're being very open about this and explaining everything," they said.

"No long-term decisions have been made," the first source concluded. The two of them are working through their differences, but "they're still taking it day by day."