A large number of things could be happening.
Wheel not set in the fork
bent fork
twisted frame
Cable issues as mentioned
troubled headset time for a repack perhaps, and check the fork out of the bike.
Sight the frame to begin with.
Does the bike track straight? line up the bike on a line, wheel it backwards steering the front wheel on the line, let the rear wheel go as it will keep the bike vertical. Watch what the rear wheel does, stay on the line or go off? The rear wheel can effect steering just a much as the front. Rear wheel could be not centered in frame, out of dish (doubtful unless you trued the wheel recently) Frame bent behind the bottom bracket.

The problem can be found, but it might take some time.

Anything preceding this or did it just begin the problem one day?