We propose a diffusion pathway for a fluorine-silicon interstitial pair $(\mathrm{F}\text{\ensuremath{-}}{\mathrm{Si}}_{\mathrm{i}})$ in silicon based on extensive first-principles density functional calculations. We find the $\mathrm{F}\text{\ensuremath{-}}{\mathrm{Si}}_{\mathrm{i}}$ pair to be most stable in the singly positive charge state under intrinsic conditions and can exist in nearly degenerate bound and unbound states. We determine that the unbound pair can undergo diffusion with a barrier of approximately $0.4--0.5\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ by a coordinated motion until dissociating into F and Si interstitials at a cost of $1.3\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$. Our results suggest that Si interstitials, when they exist in excess, can play an important role in F redistribution and precipitation during thermal treatments.