As I understand it, the primary aim of the Gallipoli campaign was to secure passage for Russian and Western Allied ships through the Sea of Marmara from the Black Sea to the Mediterranean. In comparison to the straits at Gallipoli, trying to secure passage through the Bosphorus seems almost impossible, given that the straits are surrounded by one of Europe's biggest cities, which was also the seat of power of an opposing empire.
The Allied plans for what would happen after passage through the Dardanelles were secured were somewhat nebulous, and poorly thought through. In general, the assumption was that the mere arrival of the Allied fleet off Istanbul would lead to a revolution, a change in government or, at least, force the current government to make peace. As such, there would be no need for further military action to secure passage through the Bosphorus. Should the arrival of the fleet fail to achieve these goals, then the orders for the Admiral commanding the fleet, de Roebeck, called for him to destroy the Ottoman fleet, before cutting the transport links between the European and Asian sides. After this, he was to force the Bosphorus, in combination with the Russian fleet. This would likely be an easier task than forcing the Dardanelles, as the Bosphorus forts were set up to prevent ships sailing from the Black Sea into the Sea of Marmora, not the other way round. They would also be under attack from both sides, from the Russians in the Black Sea, and from the Allied fleet in the Sea of Marmora. Once this had been achieved, de Roebeck was to move into a position to bombard Istanbul, and call on the city to surrender. Should this not occur, then de Roebeck might well have been forced to retreat back down the Dardanelles to rearm. However, there is some evidence that the arrival of the fleet would have brought peace. The American ambassador in Istanbul claimed 'Had the Allied fleets once passed the defences at the Strait, the administration of the Young Turks would have come to a bloody end.' Similarly, Enver Bey, Chief of Staff of the Ottoman Navy, believed that:
If the Fleet had got through at any time before the evacuation, it would have been able to control the communications of the Turkish Army on the Peninsula and it would have been necessary to make peace for that reason alone.