They could be trying. Its a hell of a lot harder with fiber optics than with the copper cables that were being used by the Soviets during the Cold War though. I mean it could be possible, but I don't know of any "weird" boats that the Russian Navy uses, which isnt to say they don't have any, I just don't know if they do or not. You can't just use a stock attack sub or boomer for it, its gotta have some specialized equipment.
The US Navy used to do it all the time though, check out the histories of the USS Halibut (SSN-587) after 1965 and USS Seawolf (SSN-575) post-1971. They were what we used to tap their cables. That is, until the Soviets figured out what was going on. Anecdotally (I've heard this from NSG Spooks with CT* ratings and a guy with an Intelligence Specialist rating who were all in the Navy at the time) because some cookie pusher at the State Department fucked up and asked a Politburo Central Committee member about something she couldn't have known about unless we were listening to them over a Submarine cable, this politician went and reported it to the KGB, which a couple of months later had the information stolen by the GRU who immediately had the Soviet Navy banging away at their cables with active sonar looking for the collection devices. The Soviets didn't bother encrypting anything sent over the cables because the Imperialist running dog pirates had no way of intercepting them, or so they had thought.
In reality, John Anthony Walker ("Johnnie Walker Red") probably found out about it and told the Soviets, as his son who was a member of the spy ring was stationed aboard the USS Simon Lake (AS-33) and the tenders were privy to things that the rest of the Pac and Lant fleets weren't, like what the individual boats were doing, where and when. When even the toilet paper order goes through the same supply system that missile and reactor parts do, really mundane shit ends up with an exceptionally high classification, and the Walkers were selling as much as they could to the GRU.
The Soviets located and then hauled one of the collectors up to the surface. The geniuses at the NSG or maybe even the NSA itself forgot to pull the riveted metal "Property of US Government" tabs off of them, at least on that one. The game was up at that point.
So much for plausible deniability eh?