Spatially resolving the inner dust cavity (or gap) of the so-called (pre-)transitional disks is a key to understanding the connection between the processes of planetary formation and disk dispersal. The disk around the Herbig star HD 139614 is of particular interest since it presents a pretransitional nature with an au-sized gap structure that is spatially resolved by mid-infrared interferometry in the dust distribution. With the aid of new near-infrared interferometric observations, we aim to characterize the 0.1-10 au region of the HD 139614 disk further and then identify viable mechanisms for the inner disk clearing. We report the first multiwavelength modeling of the interferometric data acquired on HD 139614 with the VLTI instruments PIONIER, AMBER, and MIDI, complemented by Herschel/PACS photometric measurements. We first performed a geometrical modeling of the new near-infrared interferometric data, followed by radiative transfer modeling of the complete dataset using the code RADMC3D. We confirm the presence of a gap structure in the warm μm-sized dust distribution, extending from about 2.5 au to 6 au, and constrained the properties of the inner dust component: e.g., a radially increasing dust surface density profile, and a depletion in dust of ~103 relative to the outer disk. Since self-shadowing and photoevaporation appears unlikely to be responsible for the au-sized gap of HD 139614, we thus tested if dynamical clearing could be a viable mechanism using hydrodynamical simulations to predict the structure of the gaseous disk. Indeed, a narrow au-sized gap is consistent with the expected effect of the interaction between a single giant planet and the disk. Assuming that small dust grains are well coupled to the gas, we found that an approximately 3 Mjup planet located at ~4.5 au from the star could, in less than 1 Myr, reproduce most of the aspects of the dust surface density profile, while no significant depletion (in gas) occurred in the inner disk, in contrast to the dust. However, this "dust-depleted" inner disk could be explained by the expected dust filtration by the gap and the efficient dust growth/fragmentation occurring in the inner disk regions. Our results support the hypothesis of a giant planet opening a gap and shaping the inner region of the HD 139614 disk. This makes HD 139614 an exciting candidate specifically for witnessing planet-disk interaction.