Each number in this pandiagonal order-4 magic square is a three-digit prime:

277 197 631 431
661 401 307 167
137 337 491 571
461 601 107 367

Add 30 to each cell and you get a new magic square, also made up of 16 three-digit primes:

307 227 661 461
691 431 337 197
167 367 521 601
491 631 137 397

Add 1092 to each cell in that one and you get a magic square of four-digit primes:

1399 1319 1753 1553
1783 1523 1429 1289
1259 1459 1613 1693
1583 1723 1229 1489

(Allan William Johnson Jr., “Related Magic Squares,” Journal of Recreational Mathematics 20:1 [January 1988], 26-27, via Clifford A. Pickover, The Zen of Magic Squares, Circles, and Stars, 2011.)