Russia would have likely annexed the city had the Entente win the Great War earlier and more decisively.
What would give Russia more claim to the city than say Greece?
Give Constantinople to Russia.
The fact that Russia is bigger, stronger, and wanted it. The only scenario where Greece could have won Constantinople after the war is a scenario where Russia still falls to a communist revolution and the Ottoman Empire/Turkey severely bungle things in the homestretch.
In 1908 (or so) Russia and Austria struck a backroom deal, Russia would support Austrias annexation of Bosnia and Austria would support Russias demand for Constantinople. OTL things kinda fell apart after the Russians realized that Austrias input on anything Constantinople relatd is meaningless leaving Russia with an egg on their face after Austria fulfilled its part of the agreement and got its reward.
Let's say... the Russians instead of being angry that they're not getting it handed over on a silver plate interpret the Austrian support as meaning they're not intervene (and convince Germany not to either), so Russia invades. What now?
Would the French do anythin? They're allies and France paid a lot of money to finance Russias buildup against Germany.
Could the UK do anything meaningful? It's not the 1850s anymore, logistics for Russia improved quite a lot and Britains army is very small.
But neither Britain, France, nor the population of Constantinople would want Russia to own the city (and therefore the Bosporus).
Chances are they would have screwed them out of that the same way they screwed the Italians.They did agree it as a war aim for Russian in the war
Chances are they would have screwed them out of that the same way they screwed the Italians.
After the war with Germany the usefulness of Russia to France comes to an end. If Germany is not involved in the war for Constantinople the French would have to stick with the Russians.