Local ultra-luminous infrared galaxies (ULIRGs) have been observed to host ubiquitous molecular outflows, including the most massive and powerful ever detected. These sources have also exceptionally excited global, galaxy-integrated CO ladders. A connection between outflows and molecular gas excitation has however never been established, since previous multi-J CO surveys were limited in spectral resolution and sensitivity and so they could only probe the global molecular gas conditions. In this work, we address this question using new, ground-based, sensitive heterodyne spectroscopy of multiple CO rotational lines (up to CO(7−6)) in a sample of 17 local ULIRGs. We used the Atacama Pathfinder Experiment (APEX) telescope to survey the CO(Jup ≥ 4) lines at a high signal-to-noise ratio, and complemented these data with CO(Jup ≤ 3) APEX and Atacama Large Millimeter Array (ALMA and ACA) observations presented in Montoya Arroyave et al. (2023, A&A, 673, A13). We detected a total of 74 (out of 75) CO lines, with up to six transitions per source. The resulting CO spectral line energy distributions (SLEDs) show a wide range in gas excitation, in agreement with previous studies on ULIRGs. Some CO SLEDs peak at Jup ∼ 3, 4, which we classify as “lower excitation”, while others plateau or keep increasing up to the highest-J CO transition probed, and we classify these as “higher excitation”. Our analysis includes for completeness the results of CO SLED fits performed with a single large velocity gradient component, but our main focus is the investigation of possible links between global CO excitation and the presence of broad and/or high-velocity CO spectral components that can contain outflowing gas. We discovered an increasing trend of line width as a function of Jup of the CO transition, which is significant at the 4σ level and appears to be driven by the eight sources that we classified as higher excitation. We further analyzed such higher-excitation ULIRGs, by performing a decomposition of their CO spectral profiles into multiple components, and we derived CO ladders that are clearly more excited for the spectral components characterized by higher velocities and/or velocity dispersion. Because these sources are known to host widespread molecular outflows, we favor an interpretation whereby the highly excited CO-emitting gas in ULIRGs resides in galactic-scale massive molecular outflows whose emission fills a large fraction of the beam of our APEX high-J CO observations. On the other hand, our results challenge alternative scenarios for which the high CO excitation in ULIRGs can be explained by classical component of the interstellar medium, such as photon- or X-ray dominated regions around the nuclear sources.