Although his teammate Sebastian Vettel has won the past three drivers' titles for the Austrian team, it will be Australian Webber first at the wheel as the 2013 pre-season begins in southern Spain on February 5 and 6.
Brazil's Totalrace added that German Vettel will drive on day three.
Webber, 36, also debuted Red Bull's championship winning 2012 car.
"I think - especially in the race - Vettel will still have his nose in front this year, but maybe in qualifying Webber can be there," former F1 driver Nick Heidfeld told Speed Week.
"The team probably put more into Vettel, as they know now very well that he can deliver the title, and also it's likely that Sebastian is going to stay longer than Webber," he added.