Diffusion and flow matching models have recently been used to solve various linear inverse problems in image restoration, such as super-resolution and inpainting. Using a pre-trained diffusion or flow-matching model as a prior, most existing methods modify the reverse-time sampling process by incorporating the likelihood information from the measurement. However, they struggle in challenging scenarios, such as high measurement noise or severe ill-posedness. In this paper, we propose Flow with Interpolant Guidance (FIG), an algorithm where reverse-time sampling is efficiently guided with measurement interpolants through theoretically justified schemes. Experimentally, we demonstrate that FIG efficiently produces highly competitive results on a variety of linear image reconstruction tasks on natural image datasets, especially for challenging tasks. Our code is available at: https://riccizz.github.io/FIG/.