  Binary supermassive black hole (SMBH) systems result from galaxy mergers, and
will eventually coalesce due to gravitational wave (GW) emission if the binary
separation can be reduced to $\lesssim 0.1$ pc by other mechanisms. Here, we
explore a gravitational self-lensing binary SMBH model for the sharp (duration
$\sim 1$ hr), quasi-regular X-ray flares -- dubbed quasi-periodic eruptions --
recently observed from two low mass active galactic nuclei: GSN 069 and RX
J1301.9+2747. In our model, the binary is observed $\sim$edge-on, such that
each SMBH gravitationally lenses light from the accretion disc surrounding the
other SMBH twice per orbital period. The model can reproduce the flare spacings
if the current eccentricity of RX J1301.9+2747 is $\epsilon_0 \gtrsim 0.16$,
implying a merger within $\sim 1000$ yrs. However, we cannot reproduce the
observed flare profiles with our current calculations. Model flares with the
correct amplitude are $\sim 2/5$ the observed duration, and model flares with
the correct duration are $\sim 2/5$ the observed amplitude. Our modelling
yields three distinct behaviours of self-lensing binary systems that can be
searched for in current and future X-ray and optical time-domain surveys: i)
periodic lensing flares, ii) partial eclipses (caused by occultation of the
background mini-disc by the foreground mini-disc), and iii) partial eclipses
with a very sharp in-eclipse lensing flare. Discovery of such features would
constitute very strong evidence for the presence of a supermassive binary, and
monitoring of the flare spacings will provide a measurement of periastron
precession.
