  (ABRIDGED) AIMS: We searched for X-ray signatures of AGN in a sample of
star-forming, relatively early-type, nearby spiral galaxies. Such galaxies are
likely to host nuclear super-massive black holes. METHODS: For 9 sources from
the Ho et al. optical sample, we isolated 10 individual Chandra X-ray sources
closest to the optical position and calculated X-ray luminosities, Lx, X-ray
colours, and, using Halpha luminosities, star formation rates. For 4 sources
with sufficient counts, we extracted X-ray spectra and fitted standard spectral
models. We assessed the significance of adding an Fe K alpha emission Gaussian
component to a fit by means of a calibration of the standard F-test. For the
rest of the sources, we estimated values for the intrinsic hydrogen column
density, NHint, and the power-law photon index, Gamma, which can reproduce the
observed X-ray colours. RESULTS: For the nuclear sources in NGC 2782 and NGC
3310, a Fe K alpha emission-line component is included with high significance,
whilst the power-law is flat (Gamma~0 and 1.3). Both sources have high
star-formation rates, with the rate for NGC 2782 being the highest in our
sample. CONCLUSIONS: Our detection of Fe K alpha emission coming from a
central, isolated source points towards a hidden AGN in 2 out of 9 relatively
early-type, star-forming spirals. The presence of an AGN is thus likely in
massive, disturbed starbursts, although the X-ray contribution is small: For
the central source, the bolometric luminosity is a fraction of a few x 10^-5 of
the Eddington luminosity. The X-ray flux of the central source is between 10
and 20% of the total galactic flux in the 0.2-10 keV band.
