The main impetus for the exile to Elba lay with the Russian tsar, Emperor Alexander I. After the Russian occupation of Paris and the French Senate's dethronement of Napoleon, Alexander I desired to appear magnanimous to his defeated foe. In a conference with Caulaincourt on 5 April, Alexander I told the French envoy that a Bonapartist regency was unthinkable and that the Emperor needed to abdicate unconditionally. But the Russian emperor reassured Caulaincourt that "[Napoleon] will be provided for; we will give him an independent state." After Caulaincourt suggested Corsica, Sardinia, and Corfu, the pair decided on the island of Elba which was now under Austrian territory.
Russia's two main allies, Austria and Britain, were quite taken aback by Alexander I's unilateral decision. Some British quarters favored a more distant exile to India (favored by Wellington) or in England as an honored prisoner. In a letter to his mistress Wilhelmine von Sagan , Metternich wrote:
If I had been here three days sooner, [the offer of Elba] wouldn't have happened. I tell you your fine emperor has made a lot of mistakes and behaved like a schoolboy escaped from his teacher. Now the teacher has returned, things will get better.
Castlereagh initially refused to countenance the Elba exile, but eventually relented for the sake of inter-Allied unity. Napoleon himself was not thrilled by Elba and actually proposed through intermediaries to Castlereagh that the British instead offer him an exile in England. But such schemes came to naught and Elba remained a fait accompli of the Russian Emperor.
Sources
Bew, John. Castlereagh: A Life. New York: Oxford University Press, 2012.
Dwyer, Philip G. Citizen Emperor: Napoleon in Power.New Haven : Yale University Press, 2013.
Price, Munro. Napoleon: The End of Glory. New York: Oxford University Press, 2014.