The 32-year-old left back will be used as cover for Kieran Tierney and will return to the Double Treble winners after a previous seven-year spell ended a year ago.
He ripped up his contract with Saudi Arabian outfit Al-Fahya in the summer because his family couldn’t settle in the Middle East country.
And he revealed he was stunned when the Hoops got in contact with him this week - because he was expecting to move to the MLS.
Izaguirre said: “I’ll be training with Celtic on Friday and the plan is to sign a two-year contract.
“My idea was to go and sign for a club in the MLS, but I didn’t have a concrete offer then all of a sudden the people from Celtic contacted me.
“As soon as that happened I was ready to sign again for them.
“I never imagined this would happen and I thank God the doors to the club have opened up for me again.
“Hopefully, I can perform to the same levels as before and I really hope we can get into the Champions League again.
“I feel physically fit and ready to play at any time. If I was going to return to a club in Europe, then to go back to a club I love very much is just incredible for me.”