Context. Lithium-rich giant stars are rare objects. For some of them, Li enrichment exceeds the abundance of this element found in solar system meteorites, suggesting that these stars have gone through a Li enhancement process. Aims: We identified a Li-rich giant HD 107028 with A(Li) > 3.3 in a sample of evolved stars observed within the PennState Toruń Planet Search. In this work we study different enhancement scenarios and we try to identify the one responsible for Li enrichment in HD 107028. Methods: We collected high-resolution spectra with three different instruments, covering different spectral ranges. We determined stellar parameters and abundances of selected elements with both equivalent width measurements and analysis, and spectral synthesis. We also collected multi-epoch high-precision radial velocities in an attempt to detect a companion. Results: Collected data show that HD 107028 is a star at the base of the red giant branch (RGB). Except for high Li abundance, we have not identified any other anomalies in its chemical composition, and there is no indication of a low-mass or stellar companion. We exclude Li production at the luminosity function bump on the RGB as the effective temperature and luminosity suggest that the evolutionary state is much earlier than the RGB bump. We also cannot confirm the Li enhancement by contamination as we do not observe any anomalies that are associated with this scenario. Conclusions: After evaluating various scenarios of Li enhancement we conclude that the Li-overabundance of HD 107028 originates from main-sequence evolution, and may be caused by diffusion processes. Based on observations obtained with the Hobby-Eberly Telescope, which is a joint project of the University of Texas at Austin, the Pennsylvania State University, Stanford University, Ludwig-Maximilians-Universität München, and Georg-August-Universität Göttingen.Based on observations made with the Italian Telescopio Nazionale Galileo (TNG) operated on the island of La Palma by the Fundación Galileo Galilei of the INAF (Istituto Nazionale di Astrofisica) at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias.Based on observations made with the Mercator Telescope, operated on the island of La Palma by the Flemish Community, at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias.