The more I think of it, I think Monk may not demand a starting spot to re-sign with the Kings. But, I think he would need assurances that there would be an open competition and the opportunity to start, if it's best for the team. I'm hoping his loyalty to Fox and to what the Kings did for his career helps sway him to re-sign. I can't see a scenario where the Kings are better off if Monk leaves in FA.
After all, I'm sure he realizes by now that even a FA pitch from another team to a "starting spot" is not a guarantee once the actual games start and the coach need to make adjustments to the lineup.