West Ham's 4-1 defeat at the hands of Premier League champions Manchester City, saw unused subsitute Carroll storm into the tunnel before the final whistle.
His actions sparked a heated exchange with Moyes in training the following day, particularly as the striker refused to apologise for leaving the bench ahead of full-time.
However, the aforementioned media source has claimed Carroll has since apologised to his boss, and has rejoined training in a bid to feature this weekend as West Ham face Leicester on Saturday.
The 29-year-old has only made 16 appearances for the 'Irons' this season, and has suffered an injury-dogged campaign.