The crackdown came after EIPR staff met with Western ambassadors in Cairo, the rights group said on Wednesday.
Egyptian Security Forces arrested the group's executive director Gasser Abdel-Razek from his home in Maadi, a suburb of Cairo, on Thursday.
EIPR said in a tweet Thursday that the forces had taken Abdel-Razek to an unknown location.
His detention comes after Egyptian security forces detained two other EIPR employees on Wednesday.
Karim Ennarah, the head of the group's criminal justice department, was arrested Wednesday while spending vacation in Dahab, a coastal town in Southern Sinai, the group said. His whereabouts and the charges he faces are unknown.
The organization's administrative manager, Mohamed Basheer, was also arrested from his home on Sunday and questioned by State Security agencies.
The crackdown on EIPR comes after a November 3 meeting the rights group held with European ambassadors in Cairo. EIPR called the arrest an "unprecedented escalation" on Wednesday.
EIPR posted pictures on Twitter of its meeting with the German, French, Italian, Spanish, Danish, Belgian, Dutch, Finnish and Swiss ambassadors. The meeting also included British and Canadian diplomats and focused on "human rights issues as well as ways of supporting the enhancement of human rights in Egypt and globally," EIPR said.
EIPR advocates for personal and religious rights, against the death penalty and publishes investigation into violations in prisons and human rights abuses in general.
Those stances leave it vulnerable in Egypt, where in recent years the government has imprisoned and restricted the travel of thousands of dissidents, political activists, journalists, protesters and human rights defenders.
Egypt has also tightened already restrictive laws regulating the work of NGOs and the media.
The arrests "serve a heavy blow against the legitimate work of human rights defenders," Amnesty International said in a statement, adding that the charges against Basheer were "bogus terrorism charges."
The French foreign ministry also expressed its deep concern over Basheer's arrest and said France intends to continue its "frank, exacting dialogue with Egypt on human rights issues."
The Egyptian foreign ministry rejected the French statement as interference in domestic affairs and claimed that it was "an attempt to influence the investigations" in an ongoing case.
"The Egyptian Initiative for Personal Rights is registered as a company while it exercises other activities in violation of the provisions of [the] law," the spokesman of the Egyptian foreign ministry Ahmed Hafez said in a statement.
Several Egyptian media outlets, largely controlled by security agencies, have published articles coinciding with the arrests accusing EIPR and other Egyptian rights groups of treason and inflaming chaos during the pro-democracy protests in 2011. They have also accused the group of meeting foreign decision-makers to tarnish Egypt's image.