But in broadening the goals of the campaign, Israel has invited growing criticism, following the initial condemnation of the abductions by Palestinian leaders and the international community.
Palestinian officials say the sweep is boosting the Islamic militants' popularity at the expense of Western-backed Palestinian President Mahmoud Abbas. They say Israeli forces have needlessly escalated tensions, targeting sites with tenuous links to Hamas while sparking protests that have left four Palestinians dead including a teenager.
They also argue that after years of Israeli and Palestinian crackdowns, Hamas was already weak when the new sweep began.
"Israel has no real targets," said Maj. Gen. Adnan Damiri of the Palestinian security forces. "Israel is doing a show."
In Israel, some asked why Prime Minister Benjamin Netanyahu hasn't come forward with what he claims is "unequivocal" proof of Hamas involvement in the June 12 abduction of three Jewish seminary students in the West Bank.
Others warned that pushing too hard could also endanger a key Israeli interest continued security coordination with Abbas' forces.
Netanyahu has demanded that Abbas dismantle the Hamas-backed unity government he formed earlier this month.
Abbas has said Israel has presented no proof of Hamas involvement in the kidnapping and that the unity government is following his political program.
"After the Americans and Europeans supported the unity government, Israel used this abduction to weaken and embarrass the Palestinian Authority," Damiri said.
In the international community, there were first signs of concern about possible overreach by Israel.
Israeli must avoid "punishing individuals for offenses they have not personally committed," senior U.N. official Jeffrey Feltman told the Security Council Monday.
As part of the battle for international public opinion, the parents of the missing teens spoke to U.N. Human Rights Council in Geneva on Tuesday.
Rachelle Fraenkel, mother of 16-year-old Naftali, said she was grateful for the worldwide condemnation of the kidnappings, but that "much more can be done and should be done by many" to bring the teens home.
She told the council about the missing that her son plays guitar, another youth loves sports and the third is an amateur pastry chef.
She also shared the parents' anguish.
"My son texted me, he said he is on his way home. And then he is gone. Every mother's nightmare is waiting and waiting endlessly for her child to come home," said Fraenkel who, like Naftali, has dual Israeli-American citizenship.
In Israel, the fate of the teens has become an obsession, with intensive media coverage and prayer vigils. It was reminiscent of the aftermath of the capture of an Israeli soldier, Gilad Schalit, by Hamas-allied militants in Gaza in 2006. Israel traded more than 1,000 Palestinian prisoners for Schalit in 2011, dozens of whom have seen been re-arrested, some in the current sweep.
There were signs Tuesday that the West Bank campaign is winding down ahead of the Muslim fasting month of Ramadan.
Israeli media quoted military officials as saying the campaign against Hamas is being scaled back for lack of targets. Four Palestinians were detained overnight Tuesday, the lowest number since the start of the operation.
Israel's military chief, Lt. Gen. Benny Gantz, said there are no plans to halt the search, but acknowledged that "with the passage of time, fears grow." He said the working assumption is that the youths are still alive.
Asked whether Israel is winding down the operation, Defense Minister Moshe Yaalon said soldiers are "concentrated on a focused operation."
The risks of Israel's clampdown became apparent over the weekend in the West Bank city of Ramallah. Early Sunday, Israeli troops searched a downtown printing company with purported links to Hamas.
Palestinians threw stones, first at Israeli soldiers and then at a Palestinian police station. The demonstrators were angry because the Palestinian officers had remained indoors during the Israeli raid, one of the rules of security cooperation.
Such cooperation has been in place since Hamas seized the Gaza Strip from Abbas in 2007, and Abbas wanted to avoid a repeat in the West Bank.
Abbas publicly defended security cooperation even after the newest Israeli crackdown, putting him at odds with much of his public.
"We don't want security coordination," about 200 marchers chanted Monday in Ramallah, but were blocked by Palestinian riot police.
Netanyahu has blamed Hamas from the start for the kidnapping.
The top Hamas leader, Khaled Mashaal, said Monday he has no information about the kidnappers, but said they are "blessed" if they took hostages that can help press for the freedom of Palestinians held by Israel.
The raids of offices and companies with purported Hamas ties have yielded 700,000 shekels ($200,000) in cash, a pile of green Hamas flags, a few hand grenades, firecrackers, computer discs and posters glorifying Hamas militants, said Maj. Guy Inbar, an Israeli defense official.
Palestinian officials said Abbas' security forces and Israel troops had already dismantled the Hamas infrastructure over the past seven years, firing Hamas-linked mosque preachers, shutting down the group's charities and drying up the flow of donations from abroad.
Ghassan Khatib, vice president of Bir Zeit University near Ramallah, said that when Israeli forces raided the campus a few days ago, they walked away with flags, posters and other materials used by one of the student groups, the Islamic bloc, during recent student council elections.
"They are not doing anything serious at all," said Khatib, a former Palestinian Authority spokesman. "Instead, they are increasing public admiration for Hamas."
Under the Palestinian unity deal, Abbas has nominally regained control of Gaza after seven years of Hamas rule. But in reality, Hamas remains in charge.
Late Tuesday, Gaza militants fired at least four rockets into southern Israel. The army said the Iron Dome defense system intercepted two rockets, while the others fell in open areas, causing no injuries or damage.
Associated Press writer Ian Deitch in Jerusalem contributed to this report.