  We present Hubble Space Telescope (HST) imaging observations of the site of
the Type-Ia supernova SN2011fe in the nearby galaxy M101, obtained about one
year prior to the event, in a narrow band centred on the HeII 4686 \AA{}
emission line. In a "single-degenerate" progenitor scenario, the hard photon
flux from an accreting white dwarf (WD), burning hydrogen on its surface over
$\sim1$ Myr should, in principle, create a HeIII Str\"{o}mgren sphere or shell
surrounding the WD. Depending on the WD luminosity, the interstellar density,
and the velocity of an outflow from the WD, the HeIII region could appear
unresolved, extended, or as a ring, with a range of possible surface
brightnesses. We find no trace of HeII 4686 \AA{} line emission in the HST
data. Using simulations, we set $2\sigma$ upper limits on the HeII 4686 \AA{}
luminosity of $L_{\rm HeII} < 3.4 \times 10^{34}$ erg s$^{-1}$ for a point
source, corresponding to an emission region of radius $r < 1.8$ pc. The upper
limit for an extended source is $L_{\rm HeII} < 1.7 \times 10^{35}$ erg
s$^{-1}$, corresponding to an extended region with $r\sim11$ pc. The largest
detectable shell, given an interstellar-medium density of 1 cm$^{-3}$, has a
radius of $\sim6$ pc. Our results argue against the presence, within the
$\sim10^5$ yr prior to the explosion, of a supersoft X-ray source of luminosity
$L_{\rm bol} \ge 3 \times 10^{37}$ erg s$^{-1}$, or of a super-Eddington
accreting WD that produces an outflowing wind capable of producing cavities
with radii of 2-6 pc.
