Zamora, 27, had been at Upton Park since January 2004, but managed just 14 games last season, scoring one goal.
Ghana international Pantsil joined West Ham in 2006 but started just seven league games for the Hammers.
"Bobby is a talented striker who not only has a desire to score goals but always looks to involve the rest of the team with his intelligent play," added Hodgson.
"I have no doubt that Bobby will play an important part for us and his quality has allowed us to strengthen our attacking options.
"John has a versatility that will provide us with more defensive options, which is extremely important to the strength and depth we have in the squad."