Murphy, 40, took charge of the team on an interim basis in September after head coach Matt O'Connor left following a 40-6 drubbing at Exeter in their season opener.
"It is a huge honour to be asked to be head coach at a club I know so well," Murphy, who joined the club as a player in 1997, said in a statement.
The eight-time Premiership champion and double European Cup winner scored 93 tries in 322 games for Tigers before taking up a role on their coaching staff in 2013.
"This club means a lot to me, I've been here a long time and we all want to enjoy success with the Tigers," Murphy added.
Leicester are eighth in the Premiership standings after three victories from nine matches.
"We've had a challenging start and we're working hard to improve performances and results, but this is also the start of a journey for this coaching team and playing group," Murphy said.
(Reporting by Shrivathsa Sridhar in Bengaluru; Editing by Hugh Lawson)