I'll link back to an answer I wrote on indigenous people in Russia that touches on its founding, but - basically what became the modern Russian state was based in the Grand Duchy of Moscow ("Muscovy") that eventually took over the rest of the Grand Principality of Vladimir-Suzdal from whence it arose (this in turn was a descendant principality of the Kievan Rus'), and then expanded. The area around St. Petersburg wasn't conquered until 1703, when the city was founded (its younger than New York, Boston or Montreal). It was the capital (even while mostly just being log cabins and a construction site) from 1704 to 1728 (when the capital was moved back to Moscow), and then from 1732 to March 1918. Even when St. Petersburg was the political capital of Russia, Moscow still held a special status as a historic, spiritual and cultural capital of the country (as well as being the second largest city). The Kremlin complex, including its palaces, was still actively used for ceremonies and coronations. Even in religious matters, Moscow held a significant role. The Metropolitan of Moscow was a major figure in the Russian Orthodox Church after Peter the Great abolished the Patriarchate - a Moscow Patriarch was restored in 1918 and has been head of the Russian Orthodox Church since.

The Bolsheviks had decamped to Moscow for pretty decent reasons - the Germans were advancing deep into Russian territory at the time and were threatening Petrograd, which in addition to being on the Baltic is very close to Russia's borders. Before the Winter War, the Finnish border was a little over 20 miles from downtown Leningrad, and even after the war and Soviet annexations, the Finnish border is about 95 miles away (the Estonian border is about 85 miles away).

On top of this, the nature of Soviet development between 1917 and 1991 meant that the positions of the cities in terms of relative size flipped. Moscow is the biggest city in Russia, and the biggest city in the former USSR by a lot. According to the 1989 Soviet census, the population of Moscow came it at just under 9 million, compared to 5 million in then-Leningrad (Kyiv was third at 2.6 million). That's not including the surrounding Moscow oblast (region) with another 6.6 million at the time, compared to Leningrad region with 1.6 million. Economic dislocation has only exacerbated that, as Moscow grew to something over 13 million today - St. Petersburg has about 5.3 million (Kyiv about 2.9 million). Soviet development definitely prioritized Moscow as a national Soviet capital, and put Leningrad/St. Petersburg on a respectable-but-still-second-tier status similar to national capitals in other Soviet Socialist Republics.

This also lends itself to how infrastructure developed - just look at a Russian rail map, or a Russian highway map, or even Russian electricity grid maps. All roads lead to Moscow, literally.

So while I can't prove a negative, there hasn't been (as far as I know) any serious proposal to move the capital to St. Petersburg, and that's even with Putin and many members of his government being from the city. It's just too exposed, strategically speaking, and too peripheral in terms of population and infrastructure, despite being the second city.