score:3
Not my favorite type of answer, but I'm coming to a no answer as I can't find anything to suggest that circumnavigating the globe was on Peter the Great's to do list. I suspect a few reasons:
Lack of Russian expertise. Unlike the nations that had already circumnavigated the globe, the Russians badly lacked naval experience in both manpower and shipbuilding (techniques and resources). Peter the Great visited Amsterdam and learned of the resources required to build ships (and dispatched convoys to a few other places/people such as the Knight Templar on Malta) and construction of the Russian fleet would depend on outside expertise (people such as https://en.wikipedia.org/wiki/Cornelius_Kruys). Most Russian sailors were fishermen and really only proficient around coastal waters and not in the open ocean
Fleet locations and existing conflicts. Peter the Great had to contend with the Swedish in the Baltic seas (great northern wars) and the Ottoman empire in the black sea. The majority of his shipbuilding resources was dedicated to these conflicts, making it difficult to justify the resources on circumnavigating the globe. The White Sea port was really their only port they could depart from that wouldn't require travelling through hostile waters before hitting the open ocean.
Need. The number 1 reason to circumnavigate the globe was to find an alternate route to India and China and bypass the giant landmass that was Russia. Being that Russia already had land routes to India and a common border with China, the need wasn't there in the same manner. Even more so that Russia already possessed pacific ports, it was simpler to construct there than to sail around the world. Russia also possesses vast amounts of land and natural resources, which was another driving factor behind the more sea faring nations setting out.
Still looking for any reference to Peter the Great wanting to circumnavigate the globe...but this is the best I have after a day of looking.