Let $V$ be a vector space over a field $F$. If $G\!\leq\! GL(V,F)$, \emph{the central dimension of $G$} is the $F$-dimension of the vector space $V/C_V(G)$. In [DEK] and [KS], soluble linear groups in which the set $\mathcal{L}_{\operatorname{icd}}(G)$ of all proper infinite central dimensional subgroups of $G$ satisfies the minimal condition and the maximal condition, respectively, have been described. On the other hand, in [MOS], periodic locally radical linear groups in which $\mathcal{L}_{\operatorname{icd}}(G)$ satisfies one of the weak chain conditions (the weak minimal condition or the weak maximal condition) have been characterized. In this paper, we begin the study of the non-periodic case by describing locally nilpotent linear groups in which $\mathcal{L}_{\operatorname{icd}}(G)$ satisfies one of the two weak chain conditions.