We apply strong magnetic fields of H=28.5 to 43 T to suppress superconductivity (SC) in the cuprates Bi2Sr2-xLaxCuO6+delta (x=0.65, 0.40, 0.25, 0.15, and 0), and investigate the low temperature (T) normal state by 63Cu nuclear spin-lattice relaxation rate (1/T1) measurements. We find that the pseudogap (PG) phase persists deep inside the overdoped region but terminates at x approximately 0.05, which corresponds to the hole doping concentration of approximately 0.21. Beyond this critical point, the normal state is a Fermi liquid that persists as the ground state when superconductivity is removed by the magnetic field. A comparison of the superconducting state with the H-induced normal state in the x=0.40 (Tc=32 K) sample indicates that there remains substantial part of the Fermi surface even in the fully developed PG state, which suggests that the PG and SC are coexisting matters.