In Legends, He approaches Darkseid and threatens him. There is some plot line why Manhunter can't attack him and Darkseid revels in it without denying he would be a threat.
had a long, competitive rivalry and enmity where they were peers...but nothing directly shown, just implied.
I think Jonn suffers more than most from being 'near Superman' level...they give him serious props when the story calls for it, but he gets punked way to easily at times.
One of my favorite images in comics though is when Jonn was fighting Asmodel (who was on fire), and struggling, but holding him off, until Superman came to take over. Made Jonn look good in overcoming, or holding out against his weakness, but also made Superman look good.