The 29-year old is a free agent after being released by West Ham United last month and Fire boss Frank Klopas says the club is tracking his situation.
"We haven't made an offer to him, at least as far as I know," Klopas said on Tuesday.
"He's a striker, yeah. We're looking at options [up top], for sure. We talked about it early. But we haven't made an offer to Carlton Cole."
Cole kicked off his career with Chelsea in 2001 during which time he was sent on loan to Wolves, Charlton Athletic and Aston Villa, before shifting to West Ham in 2006. He has represented England on seven occasions.