A regular semigroup \(S\) is called a regular \(*\)-semigroup if there is a unary operation \(*\) which satisfies the following three conditions: (i) \(xx^*x=x\), (ii) \((x^*)^*=x\), and (iii) \((xy)^*=y^*x^*\), for any \(x,y\in S\) [\textit{T. E. Nordahl, H. E. Scheiblich}, Semigroup Forum 16, 369-377 (1978; Zbl 0408.20043)]. If there a subsemigroup \(S^*\) of a regular semigroup \(S\) and a unary operation \(*\) in \(S\) satisfying: (1) \(x^*\in S^*\cap V_{S^*}(x)\), for all \(x\in S\); (2) \((x^*)^*=x\), for all \(x\in S^*\); (3) \((x^*y)^*=y^*x^{**}\) and \((xy^*)^*=y^{**}x^*\), for all \(x,y\in S\), then \(S^*\) is called a regular \(*\)-transversal of \(S\); if (3) is replaced with \((xy)^*=y^*x^*\) for all \(x,y\in S\), then \(S^*\) is called a strongly regular \(*\)-transversal of \(S\). In this paper the author discusses the class of regular semigroups with strongly regular \(*\)-transversal, proves that these semigroups are \(P\)-regular semigroups [\textit{M. Yamada, M. K. Sen}, Mem. Fac. Sci., Shimane Univ. 21, 47-54 (1987; Zbl 0648.20061)]. Finally the author characterizes the structure of the regular semigroups with a strongly regular \(*\)-transversal (Theorem 2.1), which is a main result of the paper.