What is a prefix that means near?

I'm very much in favour of quasi- for this purpose, but near serves the purpose, as does mostly.

Quasi- used to show that something is almost, but not completely, the thing described http://dictionary.cambridge.org/dictionary/british/quasi?q=quasi

The prefix peri- may suit

enclosing, encircling, or around ⇒ pericardium, pericarp, perigon
near or adjacent ⇒ perihelion

"semi: a Latin prefix to a verb, noun, or adjective meaning 'half'".

Even though it means half, it is often used in context to mean "almost", especially in cases when it is applied to non-measurable concepts. Some examples are "semi-automatic firearm", "semi-parasitic creature", or "semi-arid climate".

I think that "pseudo-" could work in the context of "near carnivorous". Pseudo means "false" or "mock", so it would not work in all cases as "near". But "pseudo-carnivorous" for some reason sounds more natural to me than "semi-carnivorous".

I think para- comes closest, when used in the sense of resembling or similar, as in para-military, para-medic or para-legal.

Others you might consider in particular contexts are quasi- and crypto-.