Like with a lot of things you can do this but there are some variables here. Namely we don't know how big your circle is and where it is in reference to the rest of the piece.
Cutting Large Stock
Your wood is sufficient for most standard jigsaw blades, that come stock with tools that I see, but they do come larger. Here is a picture of a Bosch 10" blade.
You can even see a tool review using the blade on 5" stock. I don't know the dimensions of your circle but if the diameter was wide I'm certain you could cut with larger blades if you managed to acquire one. I would suggest getting a smaller blade than 10" for your case since you don't need that extra blade flying under the wood you are cutting (Would not make for a clean cut if the tool is running at a high RPM). The 10" is shown here to illustrate your options.
Cutting Your Stock
You should aim to get a blade at roughly an inch longer than what you are cutting. So, if you can find a minimum 3 inch blade, that is designed to cut wood, that should cover it just fine. You want to be sure the sawdust clears the path to stop the blade from overheating and possibly snapping. The blade material, carbide, hss etc. might come into play but I don't have specific advice for that unfortunately.
A coping saw would be a good fit here as well if the circle was not to far inside the piece. You could drill a hole and assemble the coping blade into the hole so you don't have to have a cut line outside the circle. Coping saw are rather cheap. Never did use one on this thick of wood and they are generally used for more delicate cuts (Which is why the blades are so thin).
Since you are using a compass attachment I don't see any potential issue as long as the blade is not too wide to prevent nice arc movement. Go slow.
Edit for the coping saw
In a general sense it is still a tool to consider. In your particular case that a rather large hole and a coping saw would not work.