In this paper the notions of uniformly upper and uniformly lower $\ell $-estimates for Banach function spaces are introduced. Further, the pair $(X,Y)$ of Banach function spaces is characterized, where $X$ and $Y$ satisfy uniformly a lower $\ell $-estimate and uniformly an upper $\ell $-estimate, respectively. The integral operator from $X$ into $Y$ of the form \[ K f(x)=\varphi (x) \int _0^x k(x,y)f(y)\psi (y)\mathrm{d}y \] is studied, where $k$, $\varphi $, $\psi $ are prescribed functions under some local integrability conditions, the kernel $k$ is non-negative and is assumed to satisfy certain additional conditions, notably one of monotone type.