Yes it can be done. Easily, no, but with skill and correct machinery it is attainable.

If the concrete sub base for the stone patio was done properly there will be a grid of rebar in the body of the concrete. Two different saw blades will be required to cut thru the body - one to cut thru the stone and one to cut thru the metal bar. One could theoretically use the stone blade for the full body cut but it is a waste of a good diamond tipped blade and most craftspeople perfer to use the right blade for the right job. After cutting with with a walk behind or a hand held saw you'll probably want to clean your cuts up with a cup grinder .

A skilled mason will understand the process. I would not recommend this as a DIY'er project .

I think by the time you try to save what is there you could have it removed and start over . I will say it requires a much larger circle to get furniture arranged and I am not sure why you want a round area instead so maybe share a bit more info. I much prefer to wotk with rectangular shaped spaces except for a fire pit area.

get some curved benches and create a circle..or some big pots..just my opinion but I like your patio as it is..

You could achieve a curved edge by planting a creeping plant like phlox subulata or wooly thyme which will spread over the stone and can be trimmed in the shape you want.

The round patio is not flagstone but a cement paver. Mega Arbel. Looks like Flagstone. It would probably be cheaper to do a new patio with those than reworking the flagstone.

https://www.google.com/search?q=mega+arbel+round+patio&tbm=isch&ved=2ahUKEwiA6MGB3abtAhURfqwKHZaRDf0Q2-cCegQIABAA&oq=mega+arbel+round+patio&gs_lcp=CgNpbWcQAzoECAAQQzoCCAA6BggAEAUQHjoGCAAQCBAeOgQIABAYUJ2xAVifpAJgwacCaARwAHgBgAG7A4gBmimSAQkyLjUuNy41LjKYAQCgAQGqAQtnd3Mtd2l6LWltZ8ABAQ&sclient=img&ei=DgrDX4CpEZH8sQWWo7boDw&bih=579&biw=1280&client=firefox-b-1-d#imgrc=vD_bU-RKloIUpM

Wow—thanks so much. This is really helpful.