Word of caution: there are no circular images. There are images where corners are transparent, so that only a circle shows.
First make sure that your layer has an "alpha channel" (menu: Layer/Transparency/Add alpha channel)(if it's grayed out, it means you already have one)
Then create a circular selection ("Ellipse select tool", the 2nd one in the toolbox). You can check the "Tool options" dialog: menu: Windows/Dockable dialogs/Tool options (top of list). If you want a true circle, check the "Fixed" option, select "Aspect ratio' and enter "1:1". Depending on what kind of marks you have, you can use the diagonal framing (default) where you click one corner, and release at the opposite diagonal corner, or the radial one (check "Expand from center" in the Tool options) where you click where you want the center of the circle and release where you want the border.
Once the selection is done, you can move it (click around the middle) or extend it (click inside, but close to the sides).
One you are have the required selection, invert the selection, so that everything is selected, except your circle: menu: Select/Invert (or Ctrl-I)
Erase the selection (menu: Edit/clear or "Delete" key). You should have your central circle left, surrounded by a checkerboard pattern.
Reduce the checkerboard to the minimum by auto-cropping the image (menu: image/autocrop image)
Export the image in a format that supports transparency, like PNG. JPEG doesn't support transparent images...
If you are going to work further on the picture, save it as XCF (Gimp native format).
First make sure that your layer has an "alpha channel" (menu: Layer/Transparency/Add alpha channel)(if it's grayed out, it means you already have one)
Then create a circular selection ("Ellipse select tool", the 2nd one in the toolbox). You can check the "Tool options" dialog: menu: Windows/Dockable dialogs/Tool options (top of list). If you want a true circle, check the "Fixed" option, select "Aspect ratio' and enter "1:1". Depending on what kind of marks you have, you can use the diagonal framing (default) where you click one corner, and release at the opposite diagonal corner, or the radial one (check "Expand from center" in the Tool options) where you click where you want the center of the circle and release where you want the border.
Once the selection is done, you can move it (click around the middle) or extend it (click inside, but close to the sides).
One you are have the required selection, invert the selection, so that everything is selected, except your circle: menu: Select/Invert (or Ctrl-I)
Erase the selection (menu: Edit/clear or "Delete" key). You should have your central circle left, surrounded by a checkerboard pattern.
Reduce the checkerboard to the minimum by auto-cropping the image (menu: image/autocrop image)
Export the image in a format that supports transparency, like PNG. JPEG doesn't support transparent images...
If you are going to work further on the picture, save it as XCF (Gimp native format).