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.
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.
A diplomatic source said the bond formed between PM Naftali Bennett and Egyptian President Abdel Fatah a-Sisi is very important to Israeli-Egyptian relations.
Hundreds of congregations will recite a prayer written by fallen IDF soldier Hadar Goldin's parents over Yom Kippur and Sukkot.
The defense minister reiterated the importance of the prisoners' return, saying that Israel will not hesitate to use military force.