The boy band, put together by Simon Cowell on The X Factor, are touring the States with US boy band Big Time Rush, and have gone down so well they could be landing their own show, the Daily Star reports.
The boys also caused a stir when they performed live in New York on The Today Show and are set to perform at the Nickelodeon Kids' Choice awards later this month.
A TV source told the paper: "We'll be having meetings with One Direction when they come back for the awards. A series is definitely being talked about."
Band member Harry Styles added that the fans' reaction in America has been overwhelming.
He said: "We still can't quite believe it all, to be honest. It's sinking in slowly."