The Merrifield-Simmons index of a graph is defined as the total number of its independent sets, including the empty set. Recently, Heuberger and Wagner [Maximizing the number of independent subsets over trees with bounded degree, J. Graph Theory, \(58 (2008) 49-68\)] investigated the problem of determining the trees with the maximum Merrifield-Simmons index among trees of restricted maximum degree. In this note, we consider the problem of determining the graphs with the maximum Merrifield-Simmons index among connected graphs of restricted minimum degree. Let \(\mathcal{G}_\delta(n)\) denote the set of connected graphs of \(n\) vertices and minimum degree \(\delta\). We first conjecture that among all graphs in \(\mathcal{G}_\delta(n)\), \(n \geq 2\delta\), the graphs with the maximum Merrifield-Simmons index are isomorphic to \(K_{\delta,n-\delta}\) or \(C_5\). Then we affirm this conjecture for the case of \(\delta = 1, 2, 3\).