Martin has enjoyed a loan spell at Derby recently and he scored two goals in 13 appearances during his time at Pride Park.
He is out of contract at Norwich in the summer and Rams boss Nigel Clough confirmed earlier this week that the club were close to agreeing a deal for the forward.
The 24-year-old started his career at Carrow Road and has scored 34 goals in 117 appearances, but has featured only five times in the Premier League since Norwich won promotion back to the Premier League in 2011.
Prior to his stint at Derby, he scored once in 12 games for Swindon Town this season and has also previously had loan spells at Crystal Palace and Luton Town.
Martin will become the second signing Derby have agreed since the end of the regular Championship campaign after goalkeeper Lee Grant accepted an offer to rejoin the club this week.