The activists started out on Wednesday from the Goldin family home in Kfar Saba and had intended to finish the event at the Erez Crossing into Gaza.
“No one in this government intends to bring my brother home,” said Tzur. “We the public will bring them home.”
The weight of the US could significantly influence the development and trajectory of the negotiations, said President Isaac Herzog.
The idea of entering Gaza, left-wing activist Yoel Marshak said, was born in part out of his relationship with Palestinian journalist Sami Obeid.
Hamas had announced on Monday that the health of one of the captives it is holding was deteriorating.
"Cynical and criminal" Hamas "holds mentally ill civilians in violation of all international conventions and laws," said the Prime Minister’s Office.
Hamas is holding Avera Mengistu, Hisham al-Sayed, St.-Sgt. Oron Shaul and Lt. Hadar Goldin.
The release of Gilad Schalit was a testament to a long-held army principle that the IDF does not abandon its soldiers on the battlefield. However, many think that the price was too high.
A Hamas delegation will reportedly discuss a possible prisoner exchange deal, among other issues, during a visit in Egypt this week.
It is widely believed that the Egyptians are leading efforts to secure a deal between Israel and Hamas.