Assuming that (i) damped Ly-α absorbers (DLAs) arise in present-day-like spiral galaxies which are immersed in isothermal dark matter halos, (ii) that these galaxies obey the Tully-Fisher σ/σ ∗ = ( L/L ∗) 1/α TF and the Holmberg R L/R ∗ = ( L/L ∗) α H relations, and (iii) that they follow the Schechter luminosity distribution, we describe how their observed number density ( dN/dz), distribution of column density ( f( N)) as well as inferred cosmological density of HI ( ωHI derived from DLA surveys are affected by gravitational lensing (GL). The 'by-pass' effect causes the lines-of-sight (LOSs) towards background QSOs to avoid the central parts of galaxies and reduces their effective cross-section for absorption; the 'amplification bias' leads observers to select QSOs whose LOSs preferentially cross galaxies close to their Einstein radius. As a consequence, the determination of the quantities dN/dz, f( N) and ωHI from DLA surveys does not only depend on the redshift z and luminosity L of galaxies responsible for the absorbers but also on the column density profile of HI within the galaxies and on the redshift zq and magnitude bq of the background QSOs. For most of the existing surveys using bq ≲ 19 QSOs, the amplification bias dominates the combined effect resulting in a slight overestimate of dN/dz, f( N) and ωHI. We mainly find that observational strategies presently used to produce high- z DLA surveys result in avoiding the signature of significant GL effects: following our model, we determine that an overestimate of ωHI by more than 10% is unlikely for the z ⋍ 1.7 existing surveys, but may reach ⋍ 35% for the low redshift ones. However, we show that, in the absence of extinction by dust and micro-lensing effects, surveys ideally designed to enhance GL effects, i.e. to search for DLAs at z ∼ 0.5 in front of very bright ( b q ⋍ 16 ), high- z ( zq > 1) QSOs, may lead 1) to overestimate by up to ⋍90% the number of DLAs per unit redshift; 2) to bias the survey towards high HI column density systems so that it could contain up to 4 times as many such systems, thus 3) to overestimate by up to ⋍170% the cosmological density of gas associated with those DLAs. Identification of the galaxies responsible for the DLAs may be severely biased towards luminous galaxies if 2/α TF - α H > 0 ; this latter effect is greatly increased for log NHI > 21 DLAs.