For two graphs $G$ and $H$, let $r(G,H)$ and $r_*(G,H)$ denote the Ramsey number and star-critical Ramsey number of $G$ versus $H$, respectively. In 1996, Li and Rousseau proved that $r(K_{m},F_{t,n})=tn(m-1)+1$ for $m\geq 3$ and sufficiently large $n$, where $F_{t,n}=K_{1}+nK_{t}$. Recently, Hao and Lin proved that $r(K_{3},F_{3,n})=6n+1$ for $n\geq 3$ and $r_{\ast}(K_{3},F_{3,n})=3n+3$ for $n\geq 4$. In this paper, we show that $r(K_{m}, sF_{t,n})=tn(m+s-2)+s$ for sufficiently large $n$ and, in particular, $r(K_{3}, sF_{t,n})=tn(s+1)+s$ for $t\in\{3,4\},n\geq t$ and $s\geq1$. We also show that $r_{\ast}(K_{3}, F_{4,n})=4n+4$ for $n\geq 4$ and establish an upper bound on $r(F_{2,m},F_{t,n})$.