Earlier this evening, Sugar tweeted what appeared to be a backhanded compliment to the US TV presenter, saying that he had to credit him with a "rare moment of glory", before a series of exchanges took place.
After sparring back and forth, Sugar posted: "@piersmorgan you NEVER answer a question. u wont state yr US viewing figs and now u wont tell me what it means to have more followers (sic)"
Morgan replied: "Since @Lord_Sugar is so weirdly obsessive about my ratings today, I think everyone should tweet him asking what HIS ratings are in U.S.
"And then tweet @Lord_Sugar asking how he feels about me about to overtake him in follower count #twittertaxiforalan
"And finally, ask @Lord_Sugar how he feels about Arsenal demolishing his beloved Spurs in the football ratings, as usual #treblefail."
When Sugar replied that his ratings were zero, because he does not appear on US television, Morgan retorted: "Never likely to be either, unless it's a documentary on Yetis."
He later added: "Anyway, must dash now. Don't go tugging what's left of that scraggly white hair out."
Morgan has faced criticism following falling ratings of his prime time show, while predecessor Larry King suggested that the programme should have received a more low-key launch.