Let \(\mathcal A\) be a unital normed algebra and let \(\mathcal M\) be a unitary Banach left \(\mathcal A\)-module. Assume that \(f:\mathcal{A}\to \mathcal{M}\) is an approximate module left derivation, i.e., it satisfies \[ \| f(x+y)-f(x)-f(y)\| \leq\theta\| x\| ^{p_1}\| y\| ^{p_2} \] and \[ \| f(xy)-x\!\cdot\!f(y)-y\cdot f(x)\| \leq\varepsilon\| x\| ^{q_1}\| y\| ^{q_2} \] for all \(x,y\in {\mathcal A}\), with some \(\theta,\varepsilon\geq 0\) and with real numbers \(p_1,p_2,q_1,q_2\) such that \(p_1+p_2,q_11\). It is proved that then \(f\) is a module left derivation, i.e., \[ f(xy)=x\cdot f(y)+y\cdot f(x),\qquad x,y\in {\mathcal A}. \] Additionally, if \({\mathcal M}={\mathcal A}\) is a semiprime unital Banach algebra and the mapping \(\mathbb R\ni t\mapsto f(tx)\in {\mathcal A}\) is continuous for each fixed \(x\in{\mathcal A}\), then \(f\) is a linear derivation into \(Z({\mathcal A})\cap \text{rad}({\mathcal A})\) where \(Z({\mathcal A})\) denotes the center of \({\mathcal A}\) and \(\text{rad}({\mathcal A})\) its Jacobson radical. In particular, if \({\mathcal A}\) is semisimple, then \(f\) has to be identically zero. The obtained results can be compared with those of \textit{I. M. Singer} and \textit{J. Wermer} [Math. Ann. 129, 260--264 (1955; Zbl 0067.35101)], \textit{M. P. Thomas} [Ann. Math. 128, 435--460 (1988; Zbl 0681.47016)] and \textit{M. Brešar} and \textit{J. Vukman} [Proc. Am. Math. Soc. 110, 7--16 (1990; Zbl 0703.16020)].