I don't recommend linseed oil or tung oil as it SEALS the wood. Generally this isn't what you want, though some basses are that way, even with varnish of other types.
My original treatment for fretboards (especially rosewood) was first a good scrubbing with pure lemon oil. It's a great cleaner and will remove all the mung that builds up there. But it's a light oil and evaporates in a month or so drying the wood. So after the lemon oil (and note you want PURE lemon oil not furniture lemon oil which has petroleum distallates in it) I would apply "bore oil". This is an oil sold for treatment of the inside of clarinets and the like. This was a pretty good, cheap and effective treatment.
These days I'm just using the "Fret Doctor" Oil which seems to work and last just a little bit better than my primitive method.
But either method does not seal the wood as tung or linseed oil does. Plus Linseed oil has a problem of turning black over time (see an old piano). Tung oil does not. Personally I prefer to keep the fretboard wood natural and unsealed. It's just a personal choice. I have some basses with poly on the fretboard. I don't like it much but they work OK.