rectangles = Rectangle /@ Tuples[Range[-8, 7, 1], {2}]

Count[rectangles, _?(Not @ RegionDisjoint[Circle[{0, 0}, Sqrt[59]], #] &)]

60

Count[rectangles, _?(RegionWithin[Disk[{0, 0}, Sqrt[59]], #] &)]

156

Update: To exclude rectangles "just touching" the circle, we can use

Count[rectangles, _?(FreeQ[_EmptyRegion | _Point]@
     RegionIntersection[Circle[{0, 0}, Sqrt[59]], #] &)]

60

Count[rectangles, _?(FreeQ[_EmptyRegion | _Point]@
     RegionIntersection[Circle[{0, 0}, Sqrt[53]], #] &)]

52

rectangles = Rectangle /@ Tuples[Range[-8, 7, 1], {2}]

Count[_?(Not @ RegionDisjoint[Circle[{0, 0}, Sqrt[59]], #] &)] @ rectangles

60

Count[_?(RegionWithin[Disk[{0, 0}, Sqrt[59]], #] &)] @ rectangles

156

Update: To exclude rectangles "just touching" the circle, we can use

Count[_?(FreeQ[_EmptyRegion | _Point]@
  RegionIntersection[Circle[{0, 0}, Sqrt[59]], #] &)] @ rectangles

60

Count[_?(FreeQ[_EmptyRegion | _Point]@
     RegionIntersection[Circle[{0, 0}, Sqrt[53]], #] &)] @ rectangles

52

rectangles = Rectangle /@ Tuples[Range[-8, 7, 1], {2}]

Count[rectangles, _?(Not @ RegionDisjoint[Circle[{0, 0}, Sqrt[59]], #] &)]

60

Count[rectangles, _?(RegionWithin[Disk[{0, 0}, Sqrt[59]], #] &)]

156

rectangles = Rectangle /@ Tuples[Range[-8, 7, 1], {2}]

Count[rectangles, _?(Not @ RegionDisjoint[Circle[{0, 0}, Sqrt[59]], #] &)]

60

Count[rectangles, _?(RegionWithin[Disk[{0, 0}, Sqrt[59]], #] &)]

156

Update: To exclude rectangles "just touching" the circle, we can use

Count[rectangles, _?(FreeQ[_EmptyRegion | _Point]@
     RegionIntersection[Circle[{0, 0}, Sqrt[59]], #] &)]

60

Count[rectangles, _?(FreeQ[_EmptyRegion | _Point]@
     RegionIntersection[Circle[{0, 0}, Sqrt[53]], #] &)]

52