ABSTRACT We present a study of the connection between the escape fraction of Lyman-alpha ($\mathrm{Ly\, \alpha }$) and Lyman-continuum (LyC) photons within a sample of N = 152 star-forming galaxies selected from the VANDELS survey at 3.85 ≤ zspec ≤ 4.95 (〈zspec〉 = 4.36). By combining measurements of H$\, \alpha$ equivalent width $(W_{\rm {\lambda }}(\rm {H\, \alpha }))$ derived from broad-band photometry with measurements of Ly$\, \alpha$ equivalent width $(W_{\rm {\lambda }}(\rm {Ly\, \alpha }))$ from the VANDELS spectra, we individually estimate $f_{\rm {esc}}^{\rm {Ly\alpha }}$ for our full sample. In agreement with previous studies, we find a positive correlation between $W_{\rm {\lambda }}(\rm {Ly\, \alpha })$ and $f_{\rm {esc}}^{\rm {Ly\alpha }}$, with $f_{\rm {esc}}^{\rm {Ly\alpha }}$ increasing from $f_{\rm {esc}}^{\rm {Ly\alpha }}\simeq 0.04$ at $W_{\rm {\lambda }}(\rm {Ly\, \alpha })=10$ Å to $f_{\rm {esc}}^{\rm {Ly\alpha }}\simeq 0.1$ at $W_{\rm {\lambda }}(\rm {Ly\, \alpha })=25$ Å. For the first time at z ≃ 4–5, we investigate the relationship between $f_{\rm {esc}}^{\rm {Ly\alpha }}$ and $f_{\rm {esc}}^{\rm {LyC}}$ using $f_{\rm {esc}}^{\rm {LyC}}$ estimates derived using the equivalent widths of low-ionization, far-ultraviolet absorption lines in composite VANDELS spectra. Our results indicate that $f_{\rm {esc}}^{\rm {LyC}}$ rises monotonically with $f_{\rm {esc}}^{\rm {Ly\alpha }}$, following a relation of the form $f_{\rm {esc}}^{\rm {LyC}}$$\simeq 0.15^{+0.06}_{-0.04}$$f_{\rm {esc}}^{\rm {Ly\alpha }}$. Based on composite spectra of sub-samples with approximately constant Wλ(Lyα), but widely different $f_{\rm {esc}}^{\rm {Ly\alpha }}$, we demonstrate that the $f_{\rm {esc}}^{\rm {LyC}}$−$f_{\rm {esc}}^{\rm {Ly\alpha }}$correlation is not driven by a secondary correlation between $f_{\rm {esc}}^{\rm {Ly\alpha }}$ and Wλ(Lyα). The observed $f_{\rm {esc}}^{\rm {LyC}}$−$f_{\rm {esc}}^{\rm {Ly\alpha }}$ correlation is in good qualitative agreement with theoretical predictions and provides further evidence that estimates of $f_{\rm {esc}}^{\rm {LyC}}$ within the Epoch of Reionization should be based on proxies sensitive to neutral gas density/geometry and dust attenuation.