The Wall Street Journal revealed the potential release date in a profile of Eddy Cue, Apple's senior vice president of internet software and services.
It said that the launch of the so-called iTunes 11 is "expected as soon as Thursday", after being delayed a month due to engineering issues that "required parts to be rebuilt".
Apple has said that iTunes 11, announced at the iPad Mini launch in late October, will roll out before the end of the November, but has not stipulated a date.
The software more closely integrates iCloud, which the Journal said will make it 'resemble the mobile version'.
The paper noted that iTunes 11 "will allow people to stream music, movies and TV shows they have purchased from Apple from one Apple device to another via the iCloud syncing service without downloading them again".
However, Apple is not yet becoming a streaming media provider, as users will still have to download the files and pay for each one, rather than taking a monthly subscription similar to rivals Spotify or Pandora.
There has previously been suggestion that Apple is working on its own online music radio service, but Cue is said to be "proceeding cautiously' with Apple's streaming plans.
Apple tends to release new software at the start of business hours at its home base of California, so it may be that iTunes 11 launches at around 5pm or 6pm UK time today.