Young neutron stars (NSs) born in core-collapse explosions are promising candidates for the central engines of fast radio bursts (FRBs), since the first localized repeating burst FRB 121102 happens in a star forming dwarf galaxy, which is similar to the host galaxies of superluminous supernovae (SLSNe) and long gamma-ray bursts (LGRBs). However, FRB 180924 and FRB 190523 are localized to massive galaxies with low rates of star formation, compared with the host of FRB 121102. Meanwhile, the offsets between the bursts and host centers are about 4 kpc and 29 kpc for FRB 180924 and FRB 190523, respectively. These properties of hosts are similar to short gamma-ray bursts (SGRBs), which are produced by mergers of binary neutron star (BNS) or neutron star-black hole (NS-BH). Therefore, the NSs powering FRBs may be formed in BNS mergers. In this paper, we study the BNS merger rates, merger times, and predict their most likely merger locations for different types of host galaxies using population synthesis method. We find that the BNS merger channel is consistent with the recently reported offsets of FRB 180924 and FRB 190523. The offset distribution of short GRBs is well reproduced by population synthesis using galaxy model which is similar to GRB hosts. The event rate of FRBs (including non-repeating and repeating), is larger than those of BNS merger and short GRBs, which requires a large fraction of observed FRBs emitting several bursts. Using curvature radiation by bunches in NS magnetospheres, we also predict the observational properties of FRBs from BNS mergers, including the dispersion measure, and rotation measure. At late times (t ≥ 1yr), the contribution to dispersion measure and rotation measure from BNS merger ejecta could be neglected.