The Hammers are in need of defensive cover with Winston Reid, James Tomkins and James Collins all ruled out.
Now they have made a move for Dutch international Heitinga, who has not played in a Premier League game for Everton this season.
New Toffees boss Roberto Martinez is happy to let Heitinga leave, with his current deal due to expire in the summer.
It is understood that West Ham are likely to take over the remaining six months on the 30-year-old's contract.
It is unlikely Everton will pick him in the squad to face Stoke City on New Year's Day ahead of his expected move to East London.
Heitinga told the Dutch media on Tuesday that he was keen to move on in order to secure a place in Holland's World Cup squad.
West Ham will also complete the signing of Peterborough striker Jaanai Gordon when the transfer window reopens on Wednesday.