In the paper the homogenization of some classes of variational problems for integral functionals defined on functions subject to pointwise oscillating constraints on the gradient is considered. It is proved that these problems can be affected by the Lavrentieff phenomenon, and that it can survive the homogenization process. More precisely, by using the \(\Gamma\)-convergence method, the asymptotic behaviour, as \(h\to+\infty\), of the following problems \(m^p_h(\Omega,\beta)=\inf\{\int_\Omega f(hx,Du) dx+\int_\Omega\beta u dx: u\in W^{1,p}(\Omega)\) \((u\in C^1(\Omega)\) if \(p=\text{``}c1\text{''})\), \(u =0\) on \(\partial\Omega\), \(|Du(x)|\leq\varphi(hx)\) for a.e. \(x\in\Omega\}\) is studied. Here \(\Omega\) is a bounded open subset of \(\mathbb{R}^n\) with Lipschitz boundary, \(\beta\in L^1(\Omega)\), \(p\in ]n,+\infty]\) or \(p=\text{``}c1\text{''}\), and \(f\) and \(\varphi\) are functions satisfying the following conditions \(f: (x,z)\in\mathbb{R}^n\times\mathbb{R}^n\to f(x,z)\in[0,+\infty[\) measurable and \(]0,1[^n-\)periodic in the \(x\) variable, convex in the \(z\) one, \(f(x,\cdot)\in L^1(]0,1[^n)\) for every \(z\in\mathbb{R}^n\), \(\varphi: x\in\mathbb{R}^n\to\varphi(x)\in[0,+\infty[\) \(]0,1[^n-\)periodic, \(\varphi\in L^q(]0,1[^n)\) with \(q\in ]n,+\infty]\), there exists \(\alpha\in\mathbb{R}_+ : \int_{]0,1[^n} f(y,\pm\sqrt n\alpha\varphi(y){\mathbf e}_j)dy<+\infty\) for every \(j\in\{1,\ldots,n\}\), where \(\{{\mathbf e}_1,\cdots,{\mathbf e}_n\}\) denotes the canonical basis of \(\mathbb{R}^n\). It is proved that \(m^p_h(\Omega,\beta)\) converges to a minimum problem of the integral type for which an explicit formula, depending on \(p\), is derived. Then, it is shown that the dependence on \(p\) can be effective, and thus that the Lavrentieff phenomenon can even survive homogenization processes.