Cooper, who grew up in Warrington, Cheshire, which is about 16 miles (26km) away, appears to say “I’m not,” before adding: “I don’t have a Scouse accent.”
The exchange happened after Chalamet picked up the Best Actor prize on Sunday for his performance in Marty Supreme.
While the Critics’ Choice Awards, held in Santa Monica, can boost momentum as the Oscars race heats up, they are regarded as being less influential than many other precursor ceremonies.
Cooper took home the prize for best supporting actor in a limited series for Adolescence.
The psychological crime drama, one of Netflix’s most-watched titles of last year, follows a 13-year-old boy arrested for the murder of a classmate.