How to create circle mesh for a cloth Sim?

Title pretty much says it. I'm trying to make a cloth sim where a circular cloth just drapes over a UV sphere. Ive tried several methods already to achieve the circle:

1. Using edit mode to transform "to sphere" the outermost vertices of a square plane, but that's hit or miss with smoothness. The faces get kind of stuck together and dont fall correctly.

2. Using a circle mesh, filling it in with a face using edit mode and then subdividing it. But this creates a really odd point in the center that doesn't really fall with the rest of the cloth.