Lets assume that Britain for whatever reason opts out of the war in 1940-41 (Halifax, worse BOB, Better Italian performance and threat against the Empire etc) AND Barbarossa starts out well for the Germans despite the lack of surprise.
Would it be possible for the Axis to pursuade Turkey to join in (say in August/September 1941)?
What would Turkey like to gain?
What could they contribute? Could they put any real real pressure on the Russian Southern front?
If naval supremacy could be established in the Black sea, what would be the axis benefits? I assume an easier fall of Sevastopol, later shipments in via Rostov. How about Novorossijsk and Sochi? Could they be used to alleviate supply line stress or is the connecting infrastructure to poor?
Would it be Possible?: Sure! If British drops out, and Barbarossa is as smashing as success as IRL, then Germany is riding high on the wave of prestige at this point. Add that to the fact that they'll likely be... shall we say, 'leaning on' Ankara for closer co-operation (Like they did with the Balkan states) and with no GB to leverage as a counterweight, even more moderate Turkish politicians may just decide they don't have a choice but to dine with the Devil, with as long of a spoon as they can manage to get in the cirumstances.
What would they want?: Firstly, to get and stay on Germany's good side (Or at least its "Nations we like enough not to attack" list). Removing Soviet pressure to get access to/bases in the Straits (and the undermining of Turkish sovergeinity and ire of the British that would entail if they conceded) would also be a good knock-on effect.
But for territorial gains? Some expansion into the Caucuses are in the cards, considering there are Turkic Muslim people's living there who could be integrated into the country with some work. Azerbaijan, with the oil field at Baku in particular, would be attractive, alongside Grozny. Territorial rectification in Western Thrace (via a Vienna Awards type situation with Bulgaria) is another potential claim Germany might be able to give them and could appeal to certain Turkish politicians.
And, of course, there's the boatload of money that can be made selling chormite ore to the German war machine.
What could they contribute: Well, the chromite ore for one thing. Providing that vital raw resource will help keep the German and Italian industries running smoothly. Access through/logistical and shipping support via the Straits would allow the Italian navy to support any operations in the Black Sea, as well as allow for major shipping/resupply from ports. This would partially eliminate the need to retool the Soviet rail system, at least for forces operating in the southern theature, allowing operations there to be larger and move more quickly than they otherwise could. This allows, for example, an earlier move on Stalingrad or a lunge towards the Caucuses and the major Soviet source of fuel and refining capacity. (Of course, if Britain drops out of the war that has other effect: namely the possible resurgence of pro-Axis factions in Iran and less L-L material available for the Persian corridor anyways, meaning the Soviet logistical position in the region is already somewhat weaker)
BUt in terms of actual military operations? While a Turkish forces might be able to operate with Army Group South by being unloaded at/supplied by Black Sea ports, I can't see the Soviet-Turkish frontier being anything more than a sideshow. The roads and terrain along the border are laughably poor, meaning no large Turkish army (Especially one that needs a lot of supplies... such as anything mechanized or backed with air power or even a heavy compliment of artillery) could operate there in good order for any length of time and would suffer unacceptably high attrition rates. It may leave the Soviet forces there "Pinned" along the border; meaning they have less troops available to bolster their position in the northern Caucuses, but that's about it. Alternatively, airbases could be constructed for the Luftwaffe to conduction "Operation Pike"-like attacks on Soviet refining facilities... but Hitler wants those more or less intact so he can make use of that oil once he seizes the territory: a move that's going to look a lot more attractive now that there's more supply flexibility in the southern theatre of operations.