Can I talk you out of a jute? Not comfy when barefoot and it sounds like a beach house. Also hard to get the sand out of it.

I can not think of a reason for a round carpet. I truly can not. Always contrived..

You have no dimensions for the space and the shape of a coffee table does not dictate dimensions /shape of any rug. The whole is what matters and we can not see the whole., nor do we know the size, and what else surrounds the space.

Square rug with ends under sectional. do not go with jute. perhaps a nice indoor/outdoor rug for the area since it is a beach house.

A standard rectangular indoor/outdoor rug will work just fine. No need for square or round.

Please heed the caveats of the current advisors.
Jute is not comfy on the feet,it traps sand,and if it gets stained the blotch is impossible to eradicate.
Indoor /outdoor of good quality should serve nicely.

If you like the jute look, there are faux jute options. They will allow you to use a beater brush on your vacuum in order to fully remove the sand that will wind up in the carpet.

I also like a square option. I've never purchased a rug based upon the shape of my coffee table (I've never purchased a round rug). I believe that the shape of your room is more important than the shape of your coffee table.

A washable square would be the perfect answer . I hate the feel of real or fake jute . Sand has a way of finding its way into rugs even seating sometimes BTW

Square, but I agree jute will not feel good on your feet and will difficult to clean.

I assumed this would be a dining room. SHOCKED it's a living space. If the room were round, and the table were round I would probably go with a round carpet. That is the only time I can imagine it working.