Yes. ExifTool can copy the filesystem modified time into EXIF date fields in batch. Example:

exiftool "-FileModifyDate>DateTimeOriginal" *.jpg

That sets DateTimeOriginal from the file’s modified date. If you need other EXIF date fields too, ExifTool can write those similarly.

Another simple option is jhead, which is well suited to JPEGs:

jhead -dsft *.jpg

That sets the EXIF timestamp from the file time. If the JPEGs have no EXIF data at all, use:

jhead -mkexif -dsft *.jpg

-mkexif creates a basic EXIF header first, then -dsft copies the file time into it. Don’t use -mkexif on files that already contain EXIF, because it creates a new blank EXIF block.

A custom script could also do this, but for most users ExifTool or jhead is the easiest batch solution.