To solve conservation laws, efficient schemes such as essentially non-oscillatory (ENO) and weighted ENO (WENO) have been introduced to control the Gibbs oscillations. Based on radial basis functions (RBFs) with the classical WENO-JS weights, a new type of WENO schemes has been proposed to solve conservation laws [J. Guo et al., J. Sci. Comput., 70 (2017), pp. 551–575]. The purpose of this paper is to introduce a new formulation of conservative finite difference RBFWENO schemes to solve conservation laws. Unlike the usual method for reconstructing the flux functions, the flux function is generated directly with the conservative variables. Comparing with Guo and Jung (2017), the main advantage of this framework is that arbitrary monotone fluxes can be employed, while in Guo and Jung (2017) only smooth flux splitting can be used to reconstruct flux functions. Several 1D and 2D benchmark problems are prepared to demonstrate the good performance of the new scheme.