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Recent anomalies in decays induced by $b\to c \ell^- \barν_\ell$ transitions raise the question about such phenomena in the $D$ decays induced by $c\to (s,d)\ell^+ν_\ell$ transitions. In the experimental side, current measurements on the pure leptonic and semileptonic $D$ decays agree with the standard model predictions, such agreements can be used to constrain the new physics (NP) contributions. In this work, we extend the standard model by assuming general effective Hamiltonians describing the $c\to (s,d)\ell^+ν_\ell$ transitions including the full set of the four-fermion operators. Within the latest experimental data, we perform a minimum $χ^2$ fit of the Wilson coefficient corresponding to each operator. The results show that the Wilson coefficients of scalar operators in muon sector are at the order of ${\cal O}(10^{-2})$, and others are at the order of ${\cal O}(10^{-3})$. The lepton flavor universality could be violated by the scalar operators. We also calculate the branching fractions, the forward-backward asymmetries and polarizations of final vector mesons and leptons with the fitted Wilson coefficients of scalar and tensor operators. It is found that the pure leptonic decays are very sensitive to the scalar operators. The effects of NP on the semileptonic decays with electron are negligible, while for the semileptonic decays with muon the effects of scalar operators will show up in the forward-backward asymmetries and polarizations of muon of $D \to Pμ^+ ν_μ$. The future measurements in BESIII and Belle II experiments will help us to test effects of NP and to further test new physics models.