In this article we consider the presence of neutrino non-standard interactions (NSI) in the production and detection processes of reactor antineutrinos at the Daya Bay experiment. We report for the first time, the new constraints on the flavor non-universal and flavor universal charged-current NSI parameters, estimated using the currently released 621 days of Daya Bay data. New limits are placed assuming that the new physics effects are just inverse of each other in the production and detection processes. With this special choice of the NSI parameters, we observe a shift in the oscillation amplitude without distorting the L/E pattern of the oscillation probability. This shift in the depth of the oscillation dip can be caused by the NSI parameters as well as by theta(13), making it quite difficult to disentangle the NSI effects from the standard oscillations. We explore the correlations between the NSI parameters and theta(13) that may lead to significant deviations in the reported value of the reactor mixing angle with the help of iso-probability surface plots. Finally, we present the limits on electron, muon/tau, and flavor universal (FU) NSI couplings with and without considering the uncertainty in the normalization of the total event rates. Assuming a perfect knowledge of the event rates normalization, we find strong upper bounds similar to 0.1% for the electron and FU cases improving the present limits by one order of magnitude. However, for a conservative error of 5% in the total normalization, these constraints are relaxed by almost one order of magnitude.