Hessenthaler had expected the 36-year-old to arrive at Priestfield on-loan before the weekend, but after Paolo Di Canio was ruled out with a knee injury the Hammers pulled out of the deal.
"We won't take the interest any further," said Hessenthaler.
"He's obviously someone we would have liked to have brought in but it's very difficult at the moment, especially with the way West Ham are now that Paolo di Canio is injured.
"It's a total no-go."