With an aim to understand the effects of breakup and transfer channels on elastic scattering and fusion cross-sections in the $^{7}{\text{Li}} + ^{27}{\text{Al}}$ reaction, simultaneous measurement of elastic scattering angular distributions and fusion cross-sections have been carried out at various energies ($E_{\text{lab}} = 8.0–16.0$ MeV) around the Coulomb barrier. Optical model (OM) analysis of the elastic scattering data does not show any threshold anomaly or breakup threshold anomaly behaviour in the energy dependence of the real and imaginary parts of the OM potential. Fusion cross-section at each bombarding energy is extracted from the measured $\alpha$-particle evaporation energy spectra at backward angles by comparing with the statistical model prediction. Results on fusion cross-sections from the present measurements along with data from the literature have been compared with the coupled-channels predictions. Detailed coupled-channels calculations have been carried out to study the effect of coupling of breakup, inelastic and transfer, channels on elastic scattering and fusion. The effect of $1n$-stripping transfer coupling was found to be significant compared to that of the projectile breakup couplings in the present system.