The pair's latest row began after the Life Stories star pointed out the dwindling audience of Sugar's BBC One business show, which garnered 7.8 and 7.1 million viewers respectively for its debut episodes on Tuesday and Wednesday night.
Morgan tweeted: "@Lord_Sugar Apprentice ratings DOWN 300k on 2009 launch, beaten by EastEnders, and 292 MILLION less than my CNN audience last night. Whoops."
Sugar then hit back at the former News of the World editor by writing: "@piersmorgan I know you are a deluded ego maniac but why lie. Making it up again. I heard newspaper editor once got fired for that."
Referring to the somewhat underwhelming ratings for Piers Morgan Tonight in the US, he added: "@piersmorgan wow you might get over 300k viewers on CNN that's 0.1% of the population . Enjoy !"
The bickering continued after the premiere of episode two of The Apprentice on Wednesday night, when Morgan pointed out the flaw in Sugar's boast about the show's "great viewing figures".
He wrote: "Oh dear, Shugs....can you confirm you lost 700k ratings last night on Apprentice, 10% of your audience?? #televisiontaxiforalan.
"We know you hate 'bullsh*****s', Shugs. So stop spinning your Apprentice ratings. They fell 700k (10%) in one night, correct?"
Morgan previously clashed with Sugar over the quality of last year's Apprentice candidates.