Let X, Y be normed linear spaces, T โ L(X, Y ) be a bounded linear operator from X to Y . One wants to solve the linear problem Ax = y for x (given y โ Y ), as well as one can. When A is invertible, the unique solution is x = A -1 y. If this is not the case, one seeks an approximate solution of the form x = By, where B is an operator from Y to X. Such B is called a generalised inverse of A. Unfortunately, in general normed linear spaces, such an approximate solution depends nonlinearly on y. We introduce the concept of bounded quasi-linear generalised inverse T h of T , which contains the single-valued metric generalised inverse T M and the continuous linear projector generalised inverse T + . If X and Y are reflexive, we prove that the set of all bounded quasi-linear generalised inverses of T , denoted by G H (T ), is not empty. In the normed linear space of all bounded homogeneous operators, the best bounded quasi-linear generalised inverse T h of T is just the Moore-Penrose metric generalised inverse T M . In the case, X and Y are finite dimension spaces R n and R m , respectively, the results deduce the main result by G.R. Goldstein and J.A. Goldstein in 2000.