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Lance Fortnow, Rahul Santhanam and I have shown some nontrivial results in this direction. For example, $NP$ has $O(n^c)$ size circuits for some fixed $c$ if and only if $P^{NP[n]}$ has $O(n^k)$ size circuits for some fixed $k$. The paper has several results along these lines, so if you're interested in the generic problem of proving that $NP$ doesn't have linear size circuits, it may be a good place to get started thinking about it.
Perhaps an even better open question is: what interesting consequences can be derived from the assumption that $NP$ has $10n$ size circuits? Even with fixed leading constants like $10$, we are still stuck!