The former former Barnsley, Notts County and Stoke City boss manager, 53, will take charge at Gresty Road after the club's next two games, with Dario Gradi remaining in charge over the weekend.
Icelander Thordarson told the club's official website: "I am very pleased to be back in England and I'm very much looking forward to working at Crewe and the task we have ahead of us.
"From my previous time in England I know what this division is about and how competitive it is.
"It doesn't matter what division you are in in England, it is always competitive and I know the standard that is required to be successful.
"I have always followed Crewe's results ever since my time at Stoke and I know the results haven't been as good as they would have wanted so far this season.
"We have a relegation battle on our hands and we will have to produce over the next few months."