  This paper first makes an attempt to investigate the partial information near
optimal control of systems governed by forward-backward stochastic differential
equations with observation noise under the assumption of a convex control
domain. By Ekeland's variational principle and some basic estimates for state
processes and adjoint processes, we establish the necessary conditions for any
$\varepsilon $-near optimal control in a local form with an error order of
exact $\varepsilon ^{% \frac{1}{2}}.$ Moreover, under additional convexity
conditions on Hamiltonian function, we prove that an $\varepsilon $-maximum
condition in terms of the Hamiltonian in the integral form is sufficient for
near-optimality.
