Let $G \otimes _f H$ denote the Sierpiński product of graphs $G$ and $H$ with respect to the function $f$. The Sierpiński general position number ${\rm gp}{_{\rm S}}(G,H)$ is introduced as the cardinality of a largest general position set in $G \otimes _f H$ over all possible functions $f$. Similarly, the lower Sierpiński general position number $\underline{\rm gp}{_{\rm S}}(G,H)$ is the corresponding smallest cardinality. The concept of vertex-colinear sets is introduced. Bounds for the general position number in terms of extremal vertex-colinear sets, and bounds for the (lower) Sierpiński general position number are proved. The extremal graphs are investigated. Formulas for the (lower) Sierpiński general position number of the \SP{s} with $K_2$ as the first factor are deduced. It is proved that if $m,n\geq 2$, then ${\rm gp}{_{\rm S}}(K_m,K_n) = m(n-1)$ and that if $n\ge 2m-2$, then $\underline{\rm gp}{_{\rm S}}(K_m,K_n) = m(n-m+1)$.