  The VISTA survey of the Magellanic Clouds System (VMC) is collecting deep
$K_\mathrm{s}$--band time--series photometry of the pulsating variable stars
hosted in the system formed by the two Magellanic Clouds and the Bridge
connecting them. In this paper we have analysed a sample of 130 Large
Magellanic Cloud (LMC) Type II Cepheids (T2CEPs) found in tiles with complete
or near complete VMC observations for which identification and optical
magnitudes were obtained from the OGLE III survey. We present $J$ and
$K_\mathrm{s}$ light curves for all 130 pulsators, including 41 BL Her, 62 W
Vir (12 pW Vir) and 27 RV Tau variables. We complement our near-infrared
photometry with the $V$ magnitudes from the OGLE III survey, allowing us to
build a variety of Period-Luminosity ($PL$), Period-Luminosity-Colour ($PLC$)
and Period-Wesenheit ($PW$) relationships, including any combination of the $V,
J, K_\mathrm{s}$ filters and valid for BL Her and W Vir classes. These
relationships were calibrated in terms of the LMC distance modulus, while an
independent absolute calibration of the $PL(K_\mathrm{s})$ and the
$PW(K_\mathrm{s},V)$ was derived on the basis of distances obtained from
$Hubble Space Telescope$ parallaxes and Baade-Wesselink technique. When applied
to the LMC and to the Galactic Globular Clusters hosting T2CEPs, these
relations seem to show that: 1) the two population II standard candles RR Lyrae
and T2CEPs give results in excellent agreement with each other; 2) there is a
discrepancy of $\sim$0.1 mag between population II standard candles and
Classical Cepheids when the distances are gauged in a similar way for all the
quoted pulsators. However, given the uncertainties, this discrepancy is within
the formal 1$\sigma$ uncertainties.
