Let \(\mathbb{Z}\) be the set of integers and \(n\geq2\) be fixed. The authors consider the class of linearly independent refinable functions \[ M_{n,h}:=\left\{ m_{n,h}(\cdot-k),k\in\mathbb{Z}\right\} , \] where \(m_{n,h}(x)\) has support \(\left[ -n,n\right] ,\) is centered at the origin and satisfies the refinement equation: \[ m_{n,h}(x)=\sum_{k=n}^{n} a_{k,h}m_{n,h}(2x-k). \] Here the parameter \(h\geq2n-1\) and the coefficients are: \[ a_{k,h}=\frac{1}{2^{h}}\left[ \binom{2n}{k+n}+4(2^{h-2n+1}-1)\binom {2n-2}{k+n-1}\right] ,\qquad k=-n,\dots ,n. \] The authors construct the discrete \((dQIr),\) respectively the integral \((iQIr)\) refinable quasi-interpolatory operators of the form \(Q_{n} f(x)=\sum_{i\in\mathbb{Z}}\lambda_{i,l}(f)m_{i}(x),\) where \(m_{i} (x)=m_{n,h}(x-i)\) and \(\lambda_{i,l}\) have the expressions: \(\lambda_{i,l}(f)=\sum_{s\in L}c_{s,n}f(i-s),\) for \(dQIr\) and respectively \(\lambda_{i,l}(f)=\sum_{s\in L}c_{s,n},\) for \(iQIr;\) \(L=\left\{ -l,\dots ,l\right\} ,\) \(l\geq n-2\) and \(c_{s,n}=c_{-s,n} ,\;s=1,2,\dots l.\) Some coefficients \(c_{s,n}\) are determined imposing the exactness of \(dQIr\) (or \(iQIr)\) on the space of polynomials \(\mathbb{P}_{2n-3}\) and the rest of free coeficients are used in order to obtain a \(Q_{n}(f)\) with small max-norm. The operators with minimal max-norms are called near minimally normed (NMN). The authors give the explicit forms of NMN quasi-interpolants for \(n=3\) and \(n=4.\) Finally, some numerical results are given and the fact that these quasi-interpolants are suitable for approximation of functions having isolated discontinuities is illustrated.