Odds are very strong that the line comes from the panel itself, and that you need to RMA the monitor.
Cross checks:
(1) Use a different PC, DVD player, laptop, table, etc, to drive the monitor. Same line it's the monitor
(2) Use VGA (which is analog) instead of HDMI and DVI (both digital)
But I'd guess with 90+ confidence that's the monitor. If the monitor is out of warranty then you'll need to live with it or get a new monitor.