Their demands include more contact with relatives and an end to Israel's practice of detentions without trial, an activist said.
The strike also comes at a hard time for the Palestinian Authority, whose leader, Mahmoud Abbas, is aging and unpopular.
Protests in support of the prisoners are being held in the occupied West Bank, with youths clashing with Israeli security forces in Bethlehem.
The Arab League called on Sunday for global protection for Palestinian prisoners held by Israel, Anadolu has reported.
Palestinian rights groups have also organized protests in several cities to mark Palestinian Prisoners Day and show support for the hunger strikers, including in Jerusalem, Hebron and Ramallah.
On Monday, the Israeli Foreign Ministry complained that jailed Palestinians should not be described as "political prisoners".
There are now 6,300 Palestinian political prisoners, 500 of them in administrative detention, according to the Palestinian prisoners' rights group, Addameer.
"Prisoners are doing the same, so it's really about who cracks first", he added, noting that if more prisoners were to join the strike, the Israeli authorities may have trouble handling the large numbers.
Roughly 6,500 Palestinian prisoners are now languishing in Israeli prisons, according to Palestinian figures.
Intelligence and Transportation Minister Yisrael Katz said Tuesday that Barghouti, is now leading the hunger strike, should have been given the death penalty instead of life imprisonment.More news: What will Bill O'Reilly do now?
"They are terrorists and incarcerated murderers who are getting what they deserve and we have no reason to negotiate with them", Israeli Public Security Minister Gilad Erdan told Army Radio on Tuesday, according to media reports.
Israeli officials have dealt with hunger strikes on the part of Palestinian protestors before, however, they are clearly rattled by the scale of this one, which comes as the global community recognizes the 50 anniversary of the Israeli occupation.
Polls suggest that Barghouti, 57, is the most popular choice to replace Abbas, 82, even though he is serving five life sentences after he was convicted of being a leader of the second intifada and of directing attacks that led to the killings of Israelis.
Mr Barghouti was moved, with fellow prisoners Karim Younis and Mahmoud Abu Srour, from Hadarim prison to solitary confinement in the Jalama facility.
There are approximately 6000 Palestinian Authority Arabs in Israeli jails.
A statement released by Hamas warned the Prison Service against harming the hunger strikers. "They were brought to justice and are treated properly under global law".
Those on hunger strike have issued a list of demands, including access to phones, extended visiting rights and better medical care.
They also want Israeli authorities to install public telephones in every prison, provide air conditioners and restore kitchens.