There is no definitive information in the anime or manga on this particular question. So it MAY be possible, theoretically, to kill a Shinigami with a Death Note. However, regardless of that, in practice, it is impossible for a human to do it, for the following reasons:
You don't have their name. Humans have no way of verifying a Shinigami's true name, much less its spelling. Correct spelling using the script in which the name is "officially" written does seem to be required. Shinigami vision, which always shows the name needed to kill, always shows the "official" spelling. For example, when Misa views Light's name, it shows up not as the English word it comes from, and not as the kana approximation, but as the official kanji character, presumably because that's the way it is on his documents/how he likes to spell it, etc. Even though this spelling makes her confused as to how to pronounce it (she initially reads it as "Tsuki" in the anime). On the other hand, the names of Western characters always show up in Western script, not in Japanese approximation. The Shinigami would have their own way of spelling their names as well. According to the manga creators, each Shinigami has their "own written language", and some even use pictures (says the wiki here: deathnote.wikia.com/wiki/Shinigami). And misspellings are not allowed: after 4 misspellings the target becomes immune from that Death Note, and misspelling it knowingly more than 4 times (such as deliberate trial and error) kills the Note's owner (Rule XXXV.1). Even with the Shinigami eye deal, Shinigami names don't seem to show up, from what I've seen in the anime. So even assuming the Shinigami is stupid enough to tell you their real, full name, good luck explaining to them why you need it spelled, in their native language.
You don't have their face. You need to visualize a face to kill. Many Shinigami seem to wear masks/makeup that cover most of their faces. Who knows if that's their actual true face or just a costume? Again, good luck asking them and getting an honest answer.
No cause of death. Ok, this one is a little complicated, but here goes... The conditions of death must be "physically possible" (Rule VI.1). If an impossible cause of death is listed, or if none is specified, the cause reverts to the default: a deadly heart attack (Rule I.4). In other words, the person's blood is no longer being pumped and they quickly suffer brain death, organ failure, etc., from lack of oxygen/nutrients. But we know for a fact that it is physically impossible to kill a Shinigami this way. They are spiritual beings that are able to phase in and out of the material world and don't need to eat, so assuming they even have such an organ as a heart, lack of physical pumping from it would not kill them. Thus it is physically impossible to kill a Shinigami with the known DEFAULT cause.
So you have to SPECIFY some other cause that is actually known to kill Shinigami. The only way we know is to make them kill someone with their Note to extend another human's life (Rule XVII.1). But that won't work either. Here's why:
A. Bystander deaths are not allowed. If you make the Shinigami write down a random human's name, you
run into Rule X.2, which prevents your Death Note entry from being
the direct cause of death of anyone other than the people you
specifically listed by name.
B. Double entries are not allowed. If you try to work around that by specifying
the name of the human the Shinigami is to kill, you run into Rule
XV.1, which states that if a name is written in 2 Death Notes, only
the first entry written takes effect. So the Shinigami's attempt to
re-enter the person's name will be void, no harm no foul, the
Shinigami lives. Even if you try to force them to do a split entry with
you (allowed in Rule XXXII.1), YOU initiate the split entry, the
human becomes YOUR victim, the Shinigami's Note is NOT the cause of
death (yours is), so the Shinigami is still not responsible.
TL;DR Only someone who knows the Shinigami's true face, true name, and how their body works, can possibly kill them with a death note (or maybe even without it). That would most likely be the Shinigami king, possibly some of the higher ranking Shinigami. Your only real option is tricking the Shinigami into doing something stupid like giving up their life voluntarily or getting into big trouble with their higher-ups (they do have a death penalty, per Rule XLVI.1).
(All Rules are here: http://deathnote.wikia.com/wiki/Rules_of_the_Death_Note)
Incidentally, the same basic idea applies to the Netflix movie version of Ryuk. We don't know anything pertinent about him either. He may threaten to kill anyone who writes his name, just because he's miffed, but that doesn't mean they have the ability to actually do it. In fact, someone did write "don't trust Ryuk" on the actual parchment pages of the Death Note, so technically he should be dead, because his name was put down in the Note. So apparently it doesn't work on him.
