Oh yes, several in fact. Quite famous ones too. I even think the quest to find the North Eastern passage actually predates the North Western one, but as I am not aware of all attempts made, this might not be correct.

One of the more famous expeditions, and as far as I am aware, the first one to go North was an English expedition organized in 1553 by Merchant Adventurers of London, a "company" administered by Sebastian Cabot. It consisted of three ships, Bona Esperanza, Edward Bonaventura and the Bona Confidentia captained by Sir Hugh Willoughby and seconded by the navigator/scholar Richard Chancellor. The mission's goal was to sail around the Scandinavian peninsula and then through the Arctic ocean which they suspected was navigable and reach the Eastern shores of Asia. In this they did not succeed. The ships eventually separated and two ships with the commander Willoughby were forced to winter in an uninhabited area in the vicinity of today's city of Murmansk where the entire crew - including the commander - unfortunately perished during the winter. The third ship commanded by Richard Chancellor had managed to reach inhabited Russian regions near today's Archangelsk, from where the locals welcomed them and in fact transport was organized for Chancellor to go to Moscow where they met the Tzar Ivan (the Terrible) and established commercial relations between England and Muscovy which continued afterwards in no small capacity. Yet overall, it seems the English dropped the idea to sail by the Northern way - for some time at least. You can find documents from and about the Willoughby - Chancellor expedition here and I can recommend the book: Tudor Adventurers: The Voyage of Discovery that Transformed England by James Evans (which in my edition it is titled Merchant Adventurers. Weird that they changed it)

The idea of North and North Eastern passage didn't stop there, and it seems it had revival in the late 16th century. Cartographers belonging to the German and especially Netherlandish schools, began again circulating maps that show the Arctic as passable, influenced by arguments by Petrus Plancius. Here's a map of Arctic from 1593 by Gerard de Jode and one world map from 1594 attributed to Planicus himself, and of Arctic from 1595 by famous Gerard Meractor that all show one or multiple passageways North. And what accompanied these maps was the belief that the long periods of sunshine (which they quite understood it was the case) would melt the ice enough for the ships to pass.

In that light in 1590s the Dutch again organised expeditions going exactly North East, around Russia, lead by Willem Barentsz - whose name is probably familiar to the general audience. There were three expeditions happening consecutively in 1594, 1595 and 1596-97, none of which passed the island of Novaya Zemlya. The last one was tragic, as Barentsz ship became stranded on the island and had to winter there, and had built an impromptu hut where some managed to survive, called Het Behouden Huys. I am mentioning this trivia only because I notice now the map of Mercator I linked had the year 1595 in the title has the location of Het Behouden Huys drawn onto it, meaning the map could only be made after 1597. A common issue in dating old maps.

To get back to the expeditions, Barentsz himself died during or shortly after wintering in the third voyage, and with him the Dutch idea of going North East also ended as it had by the time became clear the North-eEastern way was too difficult even if it would ever turn out to be possible - which seemed less and less likely. Instead efforts were put into making the voyage by the route around Cape of Good Hope, which so far was more or less monoplized by the Portuguese - and lead to establishment of Dutch East India Company - VOC.

You can read more about Barentsz voyage in the work The Three Voyages Of William Barents by Beke, Charles T., Ed. which is admittedly old, but collects the primary documents, like journals and letters from Barentsz himself. It also has in the appendix a description of Henry Hudson's North Eastern voyage which you can read about if you want more as my own knowledge of the Northern voyages ends here.