Active galactic nucleus (AGN) driven outflows are believed to play an important role in regulating the growth of galaxies, mostly via negative feedback. However, their effects on their hosts are far from clear, especially for low- and moderate-luminosity Seyferts. To investigate this issue, we obtained cold molecular gas observations, traced by the CO(2-1) transition, using the NOEMA interferometer of five nearby (distances between 19 and 58Mpc) Seyfert galaxies. The resolution of ~0.3-0.8 (~30-100pc) and field of view of NOEMA allowed us to study the CO(2-1) morphology and kinematics in the nuclear regions (~100pc) and up to radial distances of ~900pc. We detected CO(2-1) emission in all five galaxies with disky or circumnuclear ring-like morphologies. We derived cold molecular gas masses on nuclear (~100pc) and circumnuclear (~650pc) scales in the range from 10^6^ to 10^7^M_{sun}_ and from 10^7^ to 10^8^M_{sun}_, respectively. In all of our galaxies, the bulk of this gas is rotating in the plane of the galaxy. However, noncircular motions are also present. In NGC 4253, NGC 4388, and NGC 7465, we can ascribe the streaming motions to the presence of a large-scale bar. In Mrk 1066 and NGC 4388, the noncircular motions in the nuclear regions are explained as outflowing material due to the interaction of the AGN wind with molecular gas in the galaxy disk. We conclude that for an unambiguous and precise interpretation of the kinematics of the cold molecular gas, we need detailed knowledge of the host galaxy (i.e., presence of bars, interactions, etc.), and also of the ionized gas kinematics and ionization cone geometry.