Two Il-38 ASW aircraft allegedly tracked a contact near Kursk into Norwegian waters and down the coast of Norway for several days before losing contact.
On the night of 18th of August 2000, 6 days after Kursk sank, the USS Memphis (SSN-691) arrived at the Haakonsvern naval base in Bergen, Norway. She had visible damage to her bow and sail, and 12 American women had quietly arrived in Bergen. The one thing they had in common is that they were married to US sailors serving on the Memphis.
-USS Memphis docked behind KNM Bergen, an Oslo-class FF. Picture taken by Russian satellite.
On direct questioning by the Russians the Norwegian embassy in Moscow confirmed the USS Memphis had docked at Haakonsvern for repairs. Allegedly part of a fence used on LA-class SSN's sail was found near Kursk.
In the days that followed both President Clinton and the Chief Director of the CIA travelled to Moscow. Later the Norwegian government would retract their previous statement and deny that the Memphis had docked at Haakonsvern.
As far as conspiracy theories go this one is one of the more believable. Numerous collisions at sea between US and Russian subs have occurred during the Cold War, but information about the accidents was kept quiet to avoid increasing tensions.
Another theory that gained some popularity in Russia is that the Peter the Great accidentally sank the Kursk with ASW weapons during a live fire demonstration.