8x8 Blocking Artifact After Compressing Gaussian Blurred Images

I've been experimenting with using Gaussian Blur on uncompressed TIFF images. The TIFF images consist of scans of printed images with a screen ruling.

I've been noticing an 8x8 blocking artifact when compressing the blurred images to JPEG's. (The reason for not using the scanner's Descreen feature is so that I can selectively blur.)

Here, a JPEG without prior blurring appears on top, and a JPEG with prior blurring appears on the bottom:

Has anyone encountered this artifact, and found a way to avoid it? Would blurring the JPEG, instead of blurring the TIFF beforehand, be preferable?