Simon Cowell -- Get Me Out of This Divorce!

Simon Cowell is upset ... very upset he's in the middle of what is becoming a bitter divorce between Andrew Silverman and Simon's soon-to-be baby mama Lauren Silverman, and he's pressuring her to settle ... sources with direct knowledge of the situation tell TMZ.

We're told Simon HATES the bad publicity, which he feels is making him look like an underhanded cad. Our sources say Simon has painted a clear picture to Lauren -- the money she'll get from Andrew is minimal because of their prenup, so any significant cash will come from Mr. X Factor himself.

We're told Lauren is nervous and doesn't want to upset Simon, so she's feeling the heat. Her problem -- she and Andrew are at odds over custody of their 7-year-old son, and she feels she may have to give too much in order to end things and appease Simon.

Although both sides have said they want to resolve the divorce amicably for the sake of their kids, we can tell you there's a lot of bad blood on both sides and it's getting really nasty.