We investigate in a unified way the reaction mechanism of Li('H, n ) Be and Li( He, p) Be at the low energies E, 2 MeV of astrophysical interest. Assuming charge independence of the nuclear reaction amplitudes and taking account of the effects from different isospin contribution, Coulomb interaction, and associated kinematical conditions properly, we constrain the upper and lower bounds of the total cross section of 'Li( H, n ) Be theoretically by using knowledge of the Li('He, p) Be reaction. It is found that the total cross section of Li( H, n ) Be at Gamow window energy E -250 keV is dominated by the near-threshold T=1 resonance whose resonance parameters are not determined at all experimentally, although the direct reaction process makes progressively important contributions at higher energies 500 keV~E, . Primordial abundance of Be calculated in the inhomogeneous and standard big-bang models by using the inferred reaction cross section for Li( H, n) Be is compared with recent results of astronomical observations. PACS number(s): 98.80.Ft, 25.55.Hp energies. It has been shown in the analyses [7] of the Li( He,p) Be reaction that the knock-on mechanism is significant. They have also shown that the cancellation assumption, which is usually employed in the DWBA method, is inadequate. Following the same line as Werby and Edwards had for the Li( He,p) Be reaction, we will concentrate on examining whether their results are still correct for the Li( H, n) Be reaction in question. Through these studies we will meet the challenge of predicting the Li( H, n) Be reaction cross section by using all the detailed information on the Li( He,p) Be reaction data.