  We present the results obtained from detailed spectral and timing studies of
extra-galactic black hole X-ray binaries LMC~X--1 and LMC~X--3, using
simultaneous observations with {\it Nuclear Spectroscopic Telescope Array
(NuSTAR)} and {\it Neil Gehrels Swift} observatories. The combined spectra in
the $0.5-30$~keV energy range, obtained between 2014 and 2019, are investigated
for both sources. We do not find any noticeable variability in $0.5-30$~keV
light curves, with $0.1-10$~Hz fractional rms estimated to be $<2$\%. No
evidence of quasi-periodic oscillations is found in the power density spectra.
The sources are found to be in the high soft state during the observations with
disc temperature $T_{\rm in}\sim 1$~keV, photon index, $\Gamma > 2.5$ and
thermal emission fraction, $f_{\rm disc}>80$\%. An Fe K$\alpha$ emission line
is detected in the spectra of LMC~X--1, though no such feature is observed in
the spectra of LMC~X--3. From the spectral modelling, the spins of the black
holes in LMC~X--1 and LMC~X--3 are estimated to be in the range of $0.92-0.95$
and $0.19-0.29$, respectively. The accretion efficiency is found to be, $\eta
\sim 0.13$ and $\eta \sim 0.04$ for LMC~X--1 and LMC~X--3, respectively.
