We explore the feasibility of a compact high-precision Hg atomic clock based on a hollow core optical fiber. We evaluate the sensitivity of the $^1S_0$-$^3P_0$ clock transition in Hg and other divalent atoms to the fiber inner core surface at non-zero temperatures. The Casimir-Polder interaction induced $^1S_0$-$^3P_0$ transition frequency shift is calculated for the atom inside the hollow capillary as a function of atomic position, capillary material, and geometric parameters. For $^{199}mathrm{Hg}$ atoms on the axis of a silica capillary with inner radius $geq 15 ,mu mathrm{m}$ and optimally chosen thickness $dsim 1 ,mu mathrm{m}$, the atom-surface interaction induced $^1S_0$-$^3P_0$ clock transition frequency shift can be kept on the level $delta u/ u_{mathrm{Hg}} sim10^{-19}$. We also estimate the atom loss and heating due to the collisions with the buffer gas, lattice intensity noise induced heating, spontaneous photon scattering, and residual birefringence induced frequency shifts.
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