It's not free, but Craftsy has a pattern for $2:
https://www.craftsy.com/knitting/pat...lacemat/208602
Actually, it looks as if many patterns for knitted doilies would work.
Here's a different one on Craftsy for $1.60:
https://www.craftsy.com/knitting/pat...acemats/207519
Here is a free doily pattern:
https://www.freepatterns.com/detail....141&cat_id=372
And here is a free pattern for flower petal placemats knit in the round:
http://whimsicalknittingdesigns.blog...lplacemat.html
And here is a free pattern from Ravelry:
http://www.ravelry.com/patterns/libr...de-up-placemat
Edit: Actually, with a little more searching on my Ravelry account (which is free to set up), I found 12 more circular knitting patterns for placemats. Here are the photos:
http://www.ravelry.com/patterns/sear...ptioned_thumbs