We propose a framework for probabilistic forecasting of dynamical systems based on generative modeling. Given observations of the system state over time, we formulate the forecasting problem as sampling from the conditional distribution of the future system state given its current state. To this end, we leverage the framework of stochastic interpolants, which facilitates the construction of generative models between an arbitrary base distribution and the target. We design a fictitious, non-physical stochastic dynamics that takes as initial condition the current system state and produces as output a sample from the target conditional distribution in finite time and without bias. This process therefore maps a point mass centered at the current state onto a probabilistic ensemble of forecasts. We prove that the drift coefficient entering the stochastic differential equation achieving this task is non-singular, and that it can be learned efficiently by quadratic regression over the time-series data. We highlight the utility of our approach on several complex, high-dimensional forecasting problems, including stochastically forced Navier-Stokes and video prediction on the KTH and CLEVRER datasets.