This work is part of the NUGA survey of CO emission in nearby active galaxies. We present observations of NGC 4569, a member of the Virgo Cluster. We analyse the molecular gas distribution and kinematics in the central region and we investigate a possible link to the strong starburst present at the nucleus. 70% of the [ 1.1±0.2] $\times$109 $M_{\odot}$ of molecular gas detected in the inner 20” is found to be concentrated within the inner 800 pc and is distributed along the large scale stellar bar seen in near-infrared observations. A hole in the CO distribution coincides with the nucleus where most of the Hα emission and blue light are emitted. The kinematics are modelled in three different ways, ranging from the purely geometrical to the ...