In Erich's Packing Center, the section on squares in a circle, you will find that for a radius-to-edge-length ratio of $25/15=5/3=1.666\dots$ the maximal number of squares you can fit completely into the circle is $5$. At least that's the best known solution, as the site says: “Only the cases $n=1$ and $n=2$ have been proved optimal.”
If you allow up to $1/5$ of the area of each square to lie outside the circle, you could start with those configurations listed on that site. But there is no guarantee for optimality, since even the original configurations come with no such guarantee. You may want to search whether some of the solutions mentioned were published in some article, and whether that article describes methods which can be adapted to your situation. But my best bet is that everything essentially boils down to trial and error, and some good intuition.