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.
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