  The high energy GeV emission of gamma-ray bursts (GRBs), detected by
\emph{Fermi}/LAT, has a significantly different morphology compared to the
lower energy MeV emission, detected by \emph{Fermi}/GBM. Though the late time
GeV emission is believed to be synchrotron radiation produced via an external
shock, this emission as early as the prompt phase is puzzling. Meaningful
connection between these two emissions can be drawn only by an accurate
description of the prompt MeV spectrum. We perform a time-resolved spectroscopy
of the GBM data of long GRBs having significant GeV emission, using a model
consisting of 2 blackbodies and a power-law. We examine in detail the evolution
of the spectral components and found that GRBs having high GeV emission (GRB
090902B and GRB 090926A) have a delayed onset of the power-law component, in
the GBM spectrum, which lingers at the later part of the prompt emission. This
behaviour mimics the flux evolution in LAT. In contrast, bright GBM GRBs with
an order of magnitude lower GeV emission (GRB 100724B and GRB 091003) show a
coupled variability of the total and the power-law flux. Further, by analyzing
the data for a set of 17 GRBs, we find a strong correlation between the
power-law fluence in the MeV and the LAT fluence (Pearson correlation: r=0.88
and Spearman correlation: $\rho=0.81$). We demonstrate that this correlation is
not influenced by the correlation between the total and the power-law fluences
at a confidence level of 2.3$\sigma$. We speculate the possible radiation
mechanisms responsible for the correlation.
