This question is coming up almost monthly now, so I’ll say the exact same thing I said last time:
A situation that could have resulted in Turkey joining the Axis is if Stalin tried to revise the Treaty of Kars by force; the Soviets signed the Treaty of Kars in 1921, giving up their claim to the Kars region in Eastern Anatolia in return for a stable border in the Caucasus at a time when the Soviets were being threatened along all of their borders and was politically isolated and militarily weak. All of the treaties signed at the time renouncing land or acknowledging the independence of former subject states were seen by the Soviet leadership (Particularly Stalin) as being of a purely temporary, tactical nature designed to buy the time necessary to secure the Bolshevik government in place and all were later broken by the Soviets, the Treaty of Kars being the sole exception.
The 1939 invasion of Finland and June 1940 invasions of the Baltic States, Bessarabia and Northern Bukovina were all to regain previous Tsarist Russian territories lost at the end of the First World War; sending troops into Artvin, Ardahan, Kars and Igdir would be entirely consistent with that policy and would have the extra benefit of strengthening the Soviet Union’s position in the Trans-Caucasus against any possible attack on that front towards Baku. If the Soviets had attacked Anatolia and seized the eastern provinces in early 1940, the Turks may have joined Barbarossa for the same reason that the Finns did, purely to regain lost land, and been unwilling to go beyond their old borders.
The Turks became an observer to the Anti-Comintern Pact in 1941, so them becoming a full member, is not out of the question. The Turks signed a Turko-German non-aggression pact at the same time. The Turks also signed a trade agreement to supply Germany with chromium for three years; a strategically important material for armaments. There were also many in the Turkish parliament in favour of siding with the Germans, including Kazim Karabekir, the Speaker of the national assembly. The Sukru Saracoglu, Turkey’s Prime Minister from June 1942 also said that Turkey was ‘entirely on Germany’s side’ in the struggle against Bolshevism and saw a German victory as inevitable. General Cakmak also favoured joining the Germans. Signing up to the Tripartite Pact is a significantly further step that probably would not appeal to them since it would reduce Turkish freedom of action and make them a satellite of Germany and Italy, but since Finland never signed the Tripartite Pact either, this does not rule out them being co-belligerents.
It is worth noting that the Soviet Union did try to overturn the Treaty of Kars in late 1945 and massed military forces on Turkey's eastern border threatening to invade. It was only pressure from the British and Americans that prevented them proceeding.