Notts County number two Dave Kevan was also linked with the job, but opted to remain at Meadow Lane.
Davis, 43, who made nearly 150 appearances for the Alex under Dario Gradi, has been in charge at Nantwich since 2003, where he won the 2006 FA Vase.
Crewe chairman John Bowler said: "I'm delighted to welcome Steve back to the club and I'm sure he will make a major contribution to the future success of Crewe Alexandra."
Manager Thordarson was delighted with his new number two, as he told BBC Radio Stoke: "He's a simple, straightforward, hard-working man and he's worked hard to gain success in the lower leagues. I'm pleased to get him and I look forward to working with him."