Let \(f(v)\) denote the smallest positive integer for which every finite simple graph on \(v\) vertices can be embedded into some strongly regular graph on at most \(f(v)\) vertices. Van H. Vu [Combinatorica 16, No. 2, 295-299 (1996; Zbl 0858.05100)] constructed the family of strongly regular graphs based on Steiner triple systems, which gives the lower bound \(f(v)\geq c'\cdot v^2\). This paper gives the construction of strongly regular graphs based on Desarguesian affine planes, such that \(f(v)\leq c\cdot v^4\). It is proved that this upper bound is exact in a class of strongly regular graphs, obtained by this construction.