When I export an ROI from a georeferenced image (ENVI's sample image, qb_boulder_msi), I get a single digit of precision for the map coordinates and many digits of precision for lat.lon:
;
; File X File Y Map X Map Y Lat Lon
4273 6498 482384.4 4154002.8 37.53283477 -117.1993811
4274 6498 482385.6 4154002.8 37.53283479 -117.1993675
4275 6498 482386.8 4154002.8 37.53283481 -117.1993539

This depends on the pixel size of the image and it's corner location. If you bring up the Cursor Location tool, you will likely only see integers and no decimal precision? If you alter the pixel size of the image, then you would likely see a different reported precision.