The authors study the two-color Ramsey numbers \(r(K_{l,m}, K_{l,n})\) for \(l\in\{ 3,4,5\}\), and \(m\approx n\), and for \(l=3\), \(m\) fixed, and \(n\) large. In the diagonal case, they improve the old upper bound on \(r(K_{l,n}, K_{l,n})\) from \textit{F. R. K. Chung} and \textit{R. L. Graham} [J. Comb. Theory, Ser. B 18, 164-169 (1975; Zbl 0298.05122)] for \(l\in\{ 3,4,5\}\), obtaining \(r(K_{3,n}, K_{3,n})\leq 8n -2\), \(r(K_{4,n}, K_{4,n})\leq 16n -2\), \(r(K_{5,n}, K_{5,n})\leq 32n -2\) for \(n\geq 9\), as well as some lower bounds, and some similar bounds for the off-diagonal cases.