Building upon the notion of Gutman index $operatorname{SGut}(G)$, Mao and Das recently introduced the Steiner Gutman index by incorporating Steiner distance for a connected graph $G$. The emph{Steiner Gutman $k$-index} $operatorname{SGut}_k(G)$ of $G$ is defined by $operatorname{SGut}_k(G)$ $=sum_{Ssubseteq V(G), |S|=k}left(prod_{vin S}deg_G(v)right) d_G(S)$, in which $d_G(S)$ is the Steiner distance of $S$ and $deg_G(v)$ is the degree of $v$ in $G$. In this paper, we derive new sharp upper and lower bounds on $operatorname{SGut}_k$, and then investigate the Nordhaus-Gaddum-type results for the parameter $operatorname{SGut}_k$. We obtain sharp upper and lower bounds of $operatorname{SGut}_k(G)+operatorname{SGut}_k(overline{G})$ and $operatorname{SGut}_k(G)cdot operatorname{SGut}_k(overline{G})$ for a connected graph $G$ of order $n$, $m$ edges and maximum degree $Delta$, minimum degree $delta$.
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