The classic case, regularly citied in histories of the 19th C, is Alexis de Tocqueville's Democracy in America, published 1835. To quote a highly relevant passage:

The time will therefore come when one hundred and fifty millions of men will be living in North America, equal in condition, the progeny of one race, owing their origin to the same cause, and preserving the same civilization, the same language, the same religion, the same habits, the same manners, and imbued with the same opinions, propagated under the same forms. The rest is uncertain, but this is certain; and it is a fact new to the world — a fact fraught with such portentous consequences as to baffle the efforts even of the imagination.

There are, at the present time, two great nations in the world which seem to tend towards the same end, although they started from different points: I allude to the Russians and the Americans. Both of them have grown up unnoticed; and whilst the attention of mankind was directed elsewhere, they have suddenly assumed a most prominent place amongst the nations; and the world learned their existence and their greatness at almost the same time.

The second paragraph is particularly interesting in that it groups to US with Russia as the two 'great nations' of the future. This echoes an even earlier prediction from Melchior von Grimm (great name) in 1790:

Two empires will then share all the advantages of civilisation, of the power of genius, of letters, of arts, of arms and industry: Russia on the eastern side and America on the western side. Other peoples of the nucleus will be too degraded, too debased to know otherwise than by a vague and stupid tradition what we have been. (Quoted in Paul Dukes' The Making of Russian Absolutism)

So the expectation that these young and growing 'flank powers', to borrow Paul Kennedy's term, would someday surpass old and decadent Europe was not unheard of in the early 19th C. It just took another century for these predictions to deliver.