Despite spending a reported £26 million on transfers this season, the Rams were dumped out of the Championship play-offs by Hull last week, and are not expected to extend the stay of interim head coach Darren Wassall.
The Sun reports that former Everton boss Roberto Martinez is the players' number 1 choice to take over at the iPro Stadium, after another preferred candidate, Brendan Rodgers, took the Celtic job on Friday.
Warburton previously led Brentford to fifth in the Championship, a year after leading them out of League One as runners-up.
And the 53-year-old has only enhanced his reputation at Rangers, securing the Scottish Championship title and Petrofac Cup in his first season in charge.
Derby sacked previous boss Paul Clement in February, with Wassall stepping up from his position as academy manager until the end of the campaign.