The 30-year-old has 18 months left on his contract with the Hammers but has been told by manager Gianfranco Zola that he is not part of his plans.
Quashie said: “If I could sign for Birmingham, then I couldn’t ask for any more.
“I wouldn’t hesitate to come to Birmingham because I think it will be a new start for me.
“I’m really happy with how things are going and would not change a thing.
“If I could come to Birmingham I’d come straight away. Playing for Birmingham is what I want to do.”
