Let $X=\{X_n: n\in\mathbb{N}\}$ be a linear process in which the coefficients are of the form $a_i=i^{-1}\ell(i)$ with $\ell$ being a slowly varying function at the infinity and the innovations are independent and identically distributed random variables belonging to the domain of attraction of an $α$-stable law with $α\in (1, 2]$. We will establish the asymptotic behavior of the partial sum process \[ \bigg\{\sum\limits_{n=1}^{[Nt]} X_n: t\geq 0\bigg\} \] as $N$ tends to infinity, where $[t]$ is the integer part of the non-negative number $t$.