  In this paper, we evaluate the (second) lightest mass $M_{1,2}$ of
right-handed neutrino $\nu_{R1,2}$ in grand unified theories with the type-I
seesaw mechanism that predicts an almost massless neutrino $m_{1 \, \rm or \,
3} \sim 0$. By chiral perturbative treatment, the masses $M_{1,2}$ are
expressed as $M_{1} = m_{D1}^{2}/m_{11} , \, M_{2} = m_{D2}^{2} m_{11} /
(m_{11} m_{22} - m_{12}^{2})$ with the mass matrix of left-handed neutrinos $m$
in the diagonal basis of the Dirac mass matrix $m_{D}$.
  Assuming $m_{Di}$ and the unitary matrix $V$ in the singular value
decomposition $(m_{D})_{ij} = V_{ik} m_{D k} U^{\dagger}_{kj}$ are close to
observed fermion masses and the CKM matrix, $M_{1,2}$ and their allowed regions
are expressed by parameters in the low energy and unknown phases. As a result,
for $m_{D1} \simeq 0.5$ MeV and $m_{D2} \simeq 100$ MeV, we obtain $M_{1}^{\rm
NH} \simeq 3 \times 10^{4 - 6}$ GeV and $M_{2}^{\rm NH} \simeq 3 \times
10^{6-8}$ GeV in the NH, $M_{1}^{\rm IH} \simeq 5 \times 10^{3 - 4}$ GeV and
$M_{2}^{\rm IH} \simeq 4 \times 10^{8-9}$ GeV in the IH. These upper and lower
bounds are proportional to $m_{Di}^{2}$ or $m_{D1} m_{D2}$.

High Energy Physics - Phenomenology