Derby, 10th in the Championship, sacked McClaren on Sunday, five months into his second spell in charge.
Rowett has been out of work since he was sacked by Birmingham in December, when Blues were seventh in the table.
The former Burton Albion boss made 120 appearances for the Rams as a defender between 1995 and 1998.
Rowett has also been linked with the vacant Norwich job, and held talkswith Scottish Premiership side Rangers earlier this month before they appointed Pedro Caixinha.
- Birmingham City (October 2014-December 2016) – 106 games, won 42, draws 32 lost 32
- Burton Albion (March 2012-October 2014) – games 142, won 63, draws 34, lost 45