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Intensity mapping (IM) with neutral hydrogen is a promising avenue to probe the large scale structure of the Universe. In this paper, we demonstrate that using the 64-dish MeerKAT radio telescope as a connected interferometer, it is possible to make a statistical detection of HI in the post-reionization Universe. With the MIGHTEE (MeerKAT International GHz Tiered Extragalactic Exploration) survey project observing in the L-band ($856 < ν< 1712$ MHz, $z < 0.66$), we can achieve the required sensitivity to measure the HI IM power spectrum on quasi-linear scales, which will provide an important complementarity to the single-dish IM MeerKAT observations. We present a purpose-built simulation pipeline that emulates the MIGHTEE observations and forecast the constraints that can be achieved on the HI power spectrum at $z = 0.27$ for $k > 0.3$ $\rm{Mpc}^{-1}$ using the foreground avoidance method. We present the power spectrum estimates with the current simulation on the COSMOS field that includes contributions from HI, noise and point source models constructed from the observed MIGHTEE data. The results from our \textit{visibility} based pipeline are in qualitative agreement to the already available MIGHTEE data. This paper demonstrates that MeerKAT can achieve very high sensitivity to detect HI with the full MIGHTEE survey on quasi-linear scales (signal-to-noise ratio $> 7$ at $k=0.49$ $\rm{Mpc}^{-1}$) which are instrumental in probing cosmological quantities such as the spectral index of fluctuation, constraints on warm dark matter, the quasi-linear redshift space distortions and the measurement of the HI content of the Universe up to $z\sim 0.5$.