Birmingham midfielder Charlee Adams has signed a new one-year contract with the club.
Southampton are confident Jay Rodriguez will commit to a new long-term deal but the player will consider his options.
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.
Carroll scored seven goals and added four assists in 26 league appearances for the Hammers last season, having been sent out on loan with Liverpool manager Brendan Rodgers unable to accommodate the club's record signing in his system.
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.
However, Rodgers is keen to bring in funds to help his own summer spending, with Shakhtar Donetsk's £20million-rated forward Henrikh Mkhitaryan, Sporting Lisbon defender Tiago Ilori, Sevilla midfielder Luis Alberto and Sunderland goalkeeper Simon Mignolet all on the radar, with Manchester City defender Kolo Toure and Celta Vigo forward Iago Aspas already secured.
Carroll will become West Ham's third signing of the summer following Shakhtar Donetsk defender Razvan Rat and Real Betis goalkeeper Adrian.