Which arguments for $m_u \approx 0$ are still in the market? The RPP note on quarks masses has traditionally carried, and it is still there, the comment that

It is particularly important to determine the quark mass ratio mu/md,
  since there is no strong CP problem if $m_u$ =
  0.

But in recent versions they have also added some remarks about how the calculations from the MILC and RBC collaborations show that this mass is no zero.
So, does it still survive some argument for $m_u=0$? And, by the way, does the strong CP problem request exactly 0, or is it enough if it is approximately zero?
 A: The question OP is proposing is linked to the question of the mass formulas. Here, what really matters is if the mass of the u quark is indeed very near zero and if one has some compelling theoretical reason to believe this.
The strong CP problem could not be of much help here as pointed out in the Dine's review. The reason is quite simple: If one should have a $\theta$ term into QCD Lagrangian, the neutron would have a measurable electric dipole. From experiments we know that is not the case and a lower bound is fixed. But the electric dipole of the neutron does not depend only from the mass of the quark u and so, having $m_u\approx 0$ is a sufficient condition but not necessarily the right one.
From a theoretical stand point, from QCD sum rules a lower bound for the masses of u and d quarks can be estimated. The main reference is S. Narison, QCD as a Theory of Hadrons (Cambridge University Press, 2007). I report here the estimation given in this book for the sake of completeness (chapter 53 in the book):
$$(m_u+m_d)(2\ GeV)>7\ MeV.$$
This grants a small but yet finite mass and whatever mass formula should satisfy this bound. Of course, this is consistent with $m_u\approx 0$. But a more recent review (see here) gives $m_u\approx 3\ MeV$ that is not so small but it is on the strong interaction scale. Smallness of $m_u$ and $m_d$ masses makes chiral symmetry a very good yet approximate symmetry.
