Mode-division multiplexing (MDM) has attracted broad attention as it could effectively boost up transmission capability by utilizing optical modes as a spatial dimension in optical interconnects. In such a technique, different data channels are usually modulated to the respective carriers over different spatial modes by using individual parallel electro-optic modulators. Each modulated channel is then multiplexed to a multi-mode waveguide. However, the method inevitably suffers from a high cost, large device footprint and high insertion loss. Here, we design intensity and phase dual-mode modulators, enabling simultaneous modulations over two channels via a graphene-on-silicon waveguide. Our method is based on the exploration of co-planar interactions between structured graphene nanoribbons (GNs) and spatial modes in a silicon waveguide. Specifically, the zeroth-order transverse electric (TE0) and first-order transverse electric (TE1) modes are modulated separately and simultaneously by applying independent driving electrodes to different GNs in an identical modulator. Our study is expected to open an avenue to develop high-density MDM photonics integrated circuits for tera-scale optical interconnects.