It happened some time after I completed the companions questline.

The only option that appears in his dialogue box is "Follow me", which is a shame because I wanted to get him cured.

Maybe it has something to do with the fact that I've done the Purity quest with Farkas first.

Oh gosh, i did the purity quest with Farkas for a couple of hours ago, and i am thinking to continue my companions questiline.

So actually maybe i'am having the same bug if i continue with Vilkas....

--------------------------------------------------------Workaround (It should work actually)

Do the purity quest with Farkas first.(...as you did already) After this is done, when you return to Vilkas, do a regular quest and then pick the purity quest again. Vilkas will now follow you without a glitch. !!!!

I've heard of it going the other way around. People being unable to do the quest with Farkas.

Yes, there is a very old glitch involving the cure. I can't remember which one but if you do it one way the brother never actually follows you. He says he will, then you head out the door and he's no longer with you. I think I had to read the wiki on this to get it right. I wanted Vilkas as my follower long-term, because my first follower, my spouse, became a dud once we married. Vilkas was great, I crafted and enchanted nice armor and weapons for him but once I met Serana I dropped him like a hot potato and just travel Tamriel with her.

Ahhhhh..Serana - i like Serana too. lol