  Given a large-scale mixture of self-interacting dark matter (SIDM) particles
and baryon matter distributed in the early Universe, we advance here a
two-phase accretion scenario for forming supermassive black holes (SMBHs) with
masses around $\sim 10^9 M_{\odot}$ at high redshifts $z (\gsim 6)$. The first
phase is conceived to involve a rapid quasi-spherical and quasi-steady Bondi
accretion of mainly SIDM particles embedded with baryon matter onto seed black
holes (BHs) created at redshifts $z\lsim 30$ by the first generation of massive
Population III stars; this earlier phase rapidly gives birth to significantly
enlarged seed BH masses of $M_{\hbox{\tiny BH},t_1}\backsimeq 1.4\times 10^6\
M_\odot \sigma_0/(1\hbox{cm}^2\hbox{g}^{-1})(C_s/30\hbox{km s}^{-1})^4$ during
$z\sim 20-15$, where $\sigma_0$ is the cross section per unit mass of SIDM
particles and $C_s$ is the velocity dispersion in the SIDM halo referred to as
an effective "sound speed". The second phase of BH mass growth is envisaged to
proceed primarily via baryon accretion, eventually leading to SMBH masses of
$M_{\hbox{\tiny BH}}\sim 10^9 M_\odot$; such SMBHs may form either by $z\sim 6$
for a sustained accretion at the Eddington limit or later at lower $z$ for
sub-Eddington mean accretion rates. We intend to account for the reported
detections of a few SMBHs at early epochs, e.g., SDSS 1148+5251 and so forth,
without necessarily resorting to either super-Eddington baryon accretion or
very frequent BH merging processes. Only extremely massive dark SIDM halos
associated with rare peaks of density fluctuations in the early Universe may
harbour such early SMBHs or quasars. Observational consequences are discussed.
