Context. The detection of Earth-like planets, exocomets or Kuiper belts show that the different components found in the solar system should also be present in other planetary systems. Trojans are one of these components and can be considered fossils of the first stages in the life of planetary systems. Their detection in extrasolar systems would open a new scientific window to investigate formation and migration processes. Aims. In this context, the main goal of the TROY project is to detect exotrojans for the first time and to measure their occurrence rate (η-Trojan). In this first paper, we describe the goals and methodology of the project. Additionally, we used archival radial velocity data of 46 planetary systems to place upper limits on the mass of possible trojans and investigate the presence of co-orbital planets down to several tens of Earth masses. Methods. We used archival radial velocity data of 46 close-in (P < 5 days) transiting planets (without detected companions) with information from high-precision radial velocity instruments. We took advantage of the time of mid-transit and secondary eclipses (when available) to constrain the possible presence of additional objects co-orbiting the star along with the planet. This, together with a good phase coverage, breaks the degeneracy between a trojan planet signature and signals coming from additional planets or underestimated eccentricity. Results. We identify nine systems for which the archival data provide >1σ evidence for a mass imbalance between L₄ and L₅. Two of these systems provide >2σ detection, but no significant detection is found among our sample. We also report upper limits to the masses at L₄/L₅ in all studied systems and discuss the results in the context of previous findings.