The Premier League club said Monday that the 28-year-old Cole, who won three Premier League titles in a seven-year spell with Chelsea, will complete the move if he passes a medical "in the next 48 hours."
Cole is Hodgson's first signing since becoming manager on July 1 and arrives on a free transfer after his Chelsea contract ended.
Cole fills the creative midfield role vacated by Yossi Benayoun, who joined Chelsea two weeks ago.
Cole started just 14 Premier League games last season after struggling with a knee injury but recovered sufficiently to make England's World Cup squad.
But he made only two late substitute appearances in South Africa as England played poorly and was eliminated in the second round.
Cole who spent five years in West Ham's first-team squad before joining Chelsea in 2003, scored 10 goals in 56 international appearances.