In this paper we introduce the deepest and most extensive ultraviolet extragalactic source catalogs of the Virgo Cluster area to date. Archival and targeted GALEX imaging is compiled and combined to provide the deepest possible coverage over ~120 deg^2 in the NUV (lambda_eff=2316 angstroms) and ~40 deg^2 in the FUV (lambda_eff=1539 angstroms) between 180 deg <= R.A. <= 195 deg and 0 deg <= Decl. <= 20 deg. We measure the integrated photometry of 1770 extended UV sources of all galaxy types and use GALEX pipeline photometry for 1,230,855 point-like sources in the foreground, within, and behind the cluster. Extended source magnitudes are reliable to m_UV ~22, showing ~0.01 sigma difference from their asymptotic magnitudes. Point-like source magnitudes have a 1 sigma standard deviation within ~0.2 mag down to m_uv ~23. The point-like source catalog is cross-matched with large optical databases and surveys including the SDSS DR9 (> 1 million Virgo Cluster sources), the Next Generation Virgo Cluster Survey (NGVS; >13 million Virgo Cluster sources), and the NED (~30,000 sources in the Virgo Cluster). We find 69% of the entire UV point-like source catalog has a unique optical counterpart, 11% of which are stars and 129 are Virgo cluster members neither in the VCC nor part of the bright CGCG galaxy catalog (i.e., m_pg < 14.5). These data are collected in four catalogs containing the UV extended sources, the UV point-like sources, and two catalogs each containing the most relevant optical parameters of UV-optically matched point-like sources for further studies from SDSS and NGVS. The GUViCS catalogs provide a unique set of data for future works on UV and multiwavelength studies in the cluster and background environments.

35 pages, 24 figures, 15 tables, Accepted for publication in A&A