Joe did consider serving as Trump's VP.
Mika & Joe were actively involved in advising Trump during the transition period. Joe recommended that two of the show's panelists , Richard Haas & Harold Ford Jr join Trump's cabinet. Both choices were rejected by Trump. Furthermore, Joe offered to have his band perform during the inauguration balls. Trump rejected that as well. Trump did accept Mika's suggestion that Dinah Powell serve in the administration.
Joe warned Trump that he would be doomed if he did not listen to his advice.