"In this case it is hardly worth hurrying to any conclusions until the investigation determines -- as our president said -- who was behind the murder of our ambassador," Kremlin spokesman Dmitry Peskov told reporters, according to AFP.
In the first official accusation against Gulen by a top Turkish official over the killing, Turkish Foreign Minister Mevlut Cavusoglu told US counterpart John Kerry in a phone call on Tuesday that Turkey believed Gulen's group FETO was involved, the foreign ministry said.
The US-based Gulen -- who Ankara says also orchestrated an attempted coup in July -- had earlier issued a statement to condemn the assassination as a "terrorist act" that left him "shocked and deeply saddened."
Russian President Vladimir Putin will take part in mourning ceremonies for the slain ambassador on Thursday as he is laid to rest in his homeland.
Veteran diplomat Karlov was shot nine times in the back by off-duty Turkish policeman Mevlut Mert Altintas at the art gallery opening of a show of Russian photography on Monday.
His body arrived back in Moscow late Tuesday after Russian investigators landed in Turkey to help probe the assassination.
The murder stunned Ankara and Moscow, which have rowed repeatedly over the Syria conflict but had recently begun to cooperate closely on the evacuations from war-wrecked Aleppo.