The 24-year-old striker, who spent last season on loan at Upton Park from Liverpool, had been due to undergo an assessment on Wednesday.
But Hammers chairman David Gold revealed on Tuesday night that the examination was already complete.
"We managed to bring Andy's medical forward to this evening and I am delighted to inform you that he has passed with flying colours,'' Gold wrote on Twitter.
Carroll passed the tests despite a heel injury suffered towards the end of last season, ruling him out of England's friendlies against the Republic of Ireland and Brazil, and expected to sideline him for up to four months.
With the medical complete, Carroll can now complete a move worth £15million-plus, with the fee having been agreed between the two clubs weeks ago.
Liverpool will take a significant loss on the 24-year-old, who signed from Newcastle for £35million in January 2011 as a replacement for the Chelsea-bound Fernando Torres.
Carroll will become West Ham's third signing of the summer following Shakhtar Donetsk defender Razvan Rat and Real Betis goalkeeper Adrian.
WE TELL IT AS IT IS - NO BULLSHIT!
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