"Sick man of Europe" is a label given to a European country experiencing a time of economic difficulty or impoverishment. The term was first used in the mid-19th century to describe the Ottoman Empire, but has since been applied at one time or another to nearly every other major country in Europe.
John Russell in 1853, in the run up to the Crimean War, quotes Nicholas I of Russia as saying that the Ottoman Empire was "a sick man—a very sick man," a "man" who "has fallen into a state of decrepitude", or a "sick man ... gravely ill".
It is not easy to determine the actual source of the quotation. The articles cited above refer to documents held or communicated personally. The most reliable, publicly available source appears to be a book by Harold Temperley, published in 1936. Temperley gives the date for the first conversation as 9 January 1853, like Goldfrank. According to Temperley, Seymour in a private conversation had to push the Tsar to be more specific about the Ottoman Empire. Eventually, the Tsar stated, "Turkey seems to be falling to pieces, the fall will be a great misfortune. It is very important that England and Russia should come to a perfectly good understanding... and that neither should take any decisive step of which the other is not apprized." And then, closer to the attributed phrase: “We have a sick man on our hands, a man gravely ill, it will be a great misfortune if one of these days he slips through our hands, especially before the necessary arrangements are made.”
Showing the disagreement between the two countries on the Eastern Question, it is important to add that the British Ambassador G. H. Seymour agreed with Tsar Nicholas's diagnosis, but he very deferentially disagreed with the Tsar's recommended treatment of the patient; he responded, "Your Majesty is so gracious that you will allow me to make one further observation. Your Majesty says the man is sick; it is very true; but your Majesty will deign to excuse me if I remark, that it is the part of the generous and strong man to treat with gentleness the sick and feeble man."
Temperley then asserts, “The ‘sickliness’ of Turkey obsessed Nicholas during his whole reign. What he really said was omitted in the Blue Book from a mistaken sense of decorum. He said not the ‘sick man’ but the ‘bear dies…the bear is dying… you may give him musk but even musk will not long keep him alive.’”