Well, the first thing we need to do is define "population". Who precisely would be feeling Manifest Destiny? Serfs certainly did not, but at different points various parts of the Tsarist elite body, or educated intellectuals critiquing that body, suggested an idea of Siberia as the manifest destiny of the Russian Empire, explicitly analogized to the American West.

Siberia has a long and complicated history, and elite attitudes were often influenced by elite attitudes elsewhere. Early narratives emphasize conquest; after all, Siberia had the Sibir Khanate. It was not just some kind of stereotyped wasteland. The conquest is reviewed in several chronicles like the Stroganov chronicle. We are told that “everywhere indeed was the grace of God spread abroad [Throughout Siberia] and poured out abundantly", and emphasis is placed on the religious aspect.

What is wondrous about the conquest of Siberia is not that it is a new colony, but that souls have been added to the dominion of the Sovereign Tsar, and under his munificent grasp, they will be enlightened by the grace of God. It takes on a distinctly religious connotation, a conquest connotation. Siberia is a land to be conquered by any other; the bifurcation is between believer and infidel, not between civilized and uncivilized.

Siberia eventually became a hub of the fur trade. Seeking to advance their profits and feed the voracious hunger for furs in Russia and Europe, traders spread further and further east, creating portages between rivers and constructing forts at important points to maintain supply and solidify control. The actual people settling Siberia (which were quite diverse in number and kind, ranging from Cossacks to Criminals to Traders to so on and so forth) need concern us less than the impression of the metropole on the Siberian lands, however. What did Moscow (and in the 18th century St. Petersburg) have to say about Siberia?

The answer is that Siberia was Russia's gold mine. Mark Bassin quotes officials that describe Siberia as “our gold mine”, or a ‘Russian Brazil’. Here we can see the influence of European frameworks on the Russian understanding; it draws upon the idea of a colony as a major source of wealth, a resource colony. There is no reason to think that at this point in time the Russian imagination sought to describe Siberia as a particularly suitable place, beyond its ability to provide riches to Russia. Siberia in this conception is remote, exotic, an other for Russia but not the kind that necessitates the settler program.

What happened when furs declined as a stream of revenue? Siberia declined in the eyes of officials. It was no longer a gold mine but a parasite. Siberia was a place to send exiles, a place to dispatch the occasional soldier, a place not to worry about. Siberia was the other that challenged the conceptions of the Russian official that he was European. How could he be European when sitting behind his European St. Petersburg lies a Forest Primeval? It is a wasteland, where men go to rot and dreams go to die.

It is precisely the dispatch of dreamers that brings a different perspective. In the aftermath of the ill-fated 1825 Decembrist revolt, which sought to bring about a Liberal revolution of a kind in Russia, many Decembrists were sent east as a punishment. Some of them became pillars in the communities they were sent to (it helped that their noble status meant that their lives were very comfy by Siberian standards). More importantly, they still had level of freedom so long as they didn't get too antsy.

So what emerged out of this freedom was an idea of Siberia at odds with an official malaise. The Decembrists, trapped as they were in Siberia, determined that their newfound home was nothing less than a laboratory of democracy. If the American West was essential to the American Liberalism, something the Decembrists at least imagined they wanted to emulate, then Siberia was essential to Russian Liberalism. In this sense Siberia became politicized as the American West, a place from which a new rugged ideal of the democratic, equal, individual would emerge.

That didn't pan out, but the Decembrists were important for introducing the idea. It's quite possible it was discussed before, but they brought the analogy of Siberia to the West to light in a new way. In the meantime, change happened in Russia. Siberia morphed as the state did, becoming the shadow of the ambitions of each Tsar. Nicholas I's push for Official Nationality included an encouragement of Siberia as a platform for heroism. Siberia, rugged, a wasteland, a wilderness, was the place where Russian heroes were made. But as a platform for manifest destiny? Such a thing is surely a reach.

So now we come to the last three Tsars. Alexander II, Alexander III, and Nicholas II. Alexander II began the shift, but I am less familiar with him than III and especially Nicholas II. We can speak in particular of a new transformation. This was a historiographical transformation, an official transformation, and an industrial transformation. To the first, a new historiography was promoted that sought to, if not 'americanize' Siberia, then at least draw connections. Siberia was not the American West, but the American West could provide lessons for Siberia, and serve as a guidepost for the Russian future. Siberia was the jewel in Russia's crown. It would allow her to have "invincible might", as Prince Ukhtomskii said. He was a major proponent under Alexander III and Nicholas II of the "Siberian future".

Here is a revealing quote from a speech he gave regarding Russia's Imperial destiny:

This northern land of mist, forest, and ice, the extreme east of Siberia, opened up by Khabaroff, and other bold pioneers like him, the land reunited to Russia by the genius of Mooravioff-Amoorsky---is still a realm of primitive quiet, of deepest stillness and stagnation. It is only with the end of the century, with the opening up of new ways of communication with our eastern coast, that a new era with all its unforeseen consequences may begin. Meanwhile the land bears the stamp of something unformed and sad, like the life of its original settlers. All the more attention and unprejudiced judgment, then, is required of anyone who would draw a parallel between the lands of the Pacific south now opening out before us, with the emerald island of Java, the inexhaustible natural riches of the Indo-Chinese soil, the self-confident vitality of the Celestial Empire, and the marshes and retired nooks, the boundless desert borders of the country whose mission, in spite of all this, is to be a source of light for the neighboring expanse, with its countless population.

One can scarcely imagine a more prototypical declaration of Imperialism. There is the contrast of the uncivilized wilderness, the opportunities opened up by technology, the ambitions that this brings to Russia. It is a way to other, warmer climes, but it is more than that. It is the future of Russia.

One of the most fascinating examples of this is a “Guide to the Great Siberian Railway (1900)”, published by the Tsarist Ministry of Ways of Communication. It was translated into English and clearly made for at least dissemination abroad. It presents Siberia as a frontier, a place for Russians to go and be born again, a place for Russia to be elevated above the nations. It contrasts savagery, it praises pioneers, it elects to educate "Asiatic life". Russia is explicitly stated to be a colonizing power, the best colonizing power, the one above all the others.

This was the policy of Nicholas II and of Alexander III, a policy that sought to resettle peasants either forcibly or by encouragement to Siberia for their benefit and the benefit of the state. Did the peasants buy it? No. For them, they saw Siberia as an ordinary peasant might; an opportunity for land, but one that they would consider on the basis of that opportunity. The disconnect between official rhetoric and peasant reality is something that one sees often in Tsarist Russia, but is especially revealing in Siberia.

Peasants would often do many bad things that the officials didn't like. They would fraternize rather than educate with the locals, they would act quite uncivilized for what were supposed to be beacons of light, and they would just bugger off from the farms they were sent to.

There was thus some definite skepticism and criticism of the Imperial line, and manifest destiny declined as an ideology of the state after the revolution. In the Soviet period, I can't really say; I do know that Soviet historiography tended to prefer to examine Siberia through the lens of class struggle between elite and proletariat, but I am unsure about how Soviet authorities presented Siberia. There might be something to be said about Khruschev's "Virgin Lands Campaign" in this context, but it's outside of what I know.

So what do we have here? We have an idea of Manifest Destiny that draws not just on an Americanized view of the frontier but a very European bifurcation of colonial and colonized, civilized and uncivilized, and ties it into the unique mission and glory of the Russian nation. That was the vision of Manifest Destiny that was promoted in the late Tsarist administration, such as it was.