Obviously you want 2D images of the visible cube sides for making a real cube.
The projections of the circle on the sides of the cube are elliptical arcs. A competent person could calculate their radiuses and main axles. Making the projection with a 3D program is much easier than performing those calculations manually.
Even in simple freeware 3D programs you can project your circle + other details onto a cube from the wanted direction. See an example. At first the circular shape is constructed on a plane.
Unwanted parts are trimmed away and the plane is rotated to the wanted position relative to the cube. The pattern is projected onto the surface of the cube.
This can be enough for your purposes, if you do not need more 3D operations. Simply save the cube surfaces as 3D vector drawings.
I tried to etch the pattern into the cube. Here's 2 different projections of the result:
ADD: Because straight lines are straight lines also in the projection, you do not have to draw all of them in a 3D program. Most of them can be inserted manually in a drawing program if you have 2D images of the projection of a circle and its diameters from 9 to 3 and 12 to 6 a'clock.
Here it is done and opened in illustrator:
The cube is seen in isometric projection and the center of the circle is aimed to the nearest corner of the cube.