I've been reading The Russian Origins of the First World War by Sean McMeekin, in which he makes some very interesting claims about what the strategic situation prior to the July crisis looked like from the Russian perspective. It is well-known that Russia has always wanted to acquire a warm-water port, but I hadn't quite realized just how sharp this problem looked at that time:
When, in summer 1912, the Porte had briefly closed the Straits to shipping during the Italian-Turkish War, Russia's vulnerability had been painfully exposed: the volume of Black Sea exports dropped by one-third for the calendar year 1912, and revenue likewise dipped 30 percent, from 77 million pounds sterling (or nearly 800 million rubles) to 57 million (less than 600 million rubles). Heavy industry in the Ukraine, dependent on supplies imported directly through the Straits via the Black Sea, had nearly ground to a halt. Although the Straits remained open for commerce during the two Balkan wars, the general disruption to trade was already so damaging that Russia's export revenue in 1913 was still 20 percent lower than in 1911.
Accordingly, Russian plans to seize the Straits by force were made more concrete:
In fact, as Sazonov himself had informed Tsar Nicholas II in a secret telegram only two months previously, serious Russian operational planning to seize Constantinople by force dated back to 1895-1896, when they had been kicked off in hopeful response to the first major wave of Armenian uprisings and subsequent massacres. Alas, Sazonov informed the tsar, Russia's amphibious carrying capacity, in the form of warships and merchant marine vessels, was not then sufficient. Eighteen years later, it was still not quite sufficient, but not for lack of trying. The February 1914 conference may have been the first one Sazonov attended in person, but for Naval and General Staff officers, joint Straits-seizure planning conferences were a dime a dozen. Just six months previously, the Naval Staff had promised the army that the Black Sea fleet could provide enough transport ships to ferry 127,500 soldiers (including 3,500 officers), 44,000 horses, 288 guns, and 11,200 horse-drawn wagons from Odessa to Constantinople. To accomplish this feat, they would add to the existing fleet by quickly commandeering 115 civilian ships from Russia's Merchant Marine. All Black Sea port officials were already under naval command. True, it would take sixty days for all the men and war materiel to reach the Ottoman capital, but the first "echelon," comprising a bit more than a single army corps (30,000 to 50,000 men), including a full division's artillery component, could put ashore by Day 15, weather conditions permitting. By February 1914, "zero hour"-the day on which the first Russian amphibious landing forces would put ashore at the Bosphorus-had been accelerated to M + 10.
That sounds pretty concrete indeed! Amphibious operations normally take a great deal of planning and preparation – for example, Operation Overlord took a little more than a year from the decision to D-Day – and the Russians would have liked more time, but they had the problem that Turkey had no less than five dreadnought-class battleships on order, even one of which would suffice to tilt the balance of naval power in the Black Sea in Turkey's favor. (Russia could not bring in reinforcements from the Baltic, as their warships were forbidden passage through the Straits even in peacetime.) According to https://en.wikipedia.org/wiki/Re%C5%9Fadiye-class_battleship the first of these battleships was due in August 1914.
An amphibious invasion of Istanbul seems like it would be a nontrivial undertaking, but McMeeker's analysis looks solid. The Turkish defenses at that time were not nearly as well-organized as they would become under German advisors prior to the landing at Gallipoli.
And Russia had motive. Invading Turkey over land would have involved going through Romania and Bulgaria to the west (and their relationship with Bulgaria was by no means trusting enough to make it likely that free passage of Russian armies would've been allowed) or fighting hundreds of miles through the Caucasus Mountains to the east (terrain very poorly suited to rapid advance).
The Russian army was not noted for speed, but they could sometimes move fast, if they knew they had to, as in their campaign in East Prussia in the autumn of 1914 that led to the battle of Tannenberg.
Did Russia actually have the means, both material and organizational, to launch an amphibious invasion of Istanbul by August 1914?