Elias Abuelazam is "not going to be brought back today, tomorrow or Monday," Michigan State Police Lt. Stephen Sipes said. "We have 15 days to get him back. There is no hurry at the moment."
The 33-year-old waived extradition to Michigan during a Friday court hearing in Atlanta.
Abuelazam was arrested there Wednesday as he prepared to board a flight to Tel Aviv. He grew up nearby, in a small Arab Christian community in the Israeli town of Ramle.
Along with attacks in Michigan, he is a suspect in stabbings in Ohio and Virginia.
Michigan State Police and prosecutors said they're working with their counterparts in those other states.
"There are security concerns for his safety and the safety of everyone," Sipes said. "This is not a person wanted on an outstanding check warrant."
No threats have been reported against Abuelazam, but the violence associated with the attacks and his size present safety issues.
Abuelazam, who's described as being more than six feet tall and weighing more than 200 pounds, is charged with assault with intent to murder in last month's brutal stabbing of a man in Flint, about 50 miles northwest of Detroit.
He is a suspect in a total of 18 attacks. The first 14 occurred in and around Flint in Michigan's Genesee County beginning May 24 and ending early this month. They were followed by two knifings and a hammer attack in Leesburg, Va., and a stabbing in Toledo, Ohio.
"It's such a high profile case and a logistical challenge," Assistant Genesee County Prosecutor John Potbury said Saturday.
Law enforcement in three states are compiling evidence as they prepare charges in each of the attacks. That evidence may have to be shared among agencies in Michigan, Ohio and Virginia.
If - and when - Abuelazam would be transferred from state to state to face charges must also be worked out, Potbury said.
Also Saturday, a lawyer in Abuelazam's hometown of Ramle in Israel said the evidence expected to be presented in the case is circumstantial. Giora Zilbershtein said he will work with an attorney hired to represent Abuelazam in the United States.
"For the many years that he lived in the United States and worked and married, he never got into trouble with the law," Zilbershtein said.
He added, "I still need to learn the case and be in touch with his lawyer in the U.S. and build a defense. It's too early to give answers about his defense."
Abuelazam was a suspect in a screwdriver stabbing of an acquaintance in Ramle during an argument about six months ago. The victim declined to press charges.