"Throughout the talks that we have been making in Cairo, I think there might be positive results in the next few days on reaching an understanding on a truce with the Israelis," Hamas' Mousa Abul Marzouk told CBS News.
The number two figure in the militants' organization said any long-term cease-fire with Israel would include an immediate lifting of the 18-month blockade on the impoverished seaside Gaza Strip, and opening its border crossings to allow in humanitarian aid for Gaza's 1.5 million people.
Israeli and Palestinian officials have been shuttling to Cairo for talks with Egypt's intelligence chief and Middle East mediator Omar Suleiman for a cease-fire since the end of Israel's massive 22-day onslaught on Gaza, which killed at least 1,330 Palestinians and wounded some 5,000.
The fighting ended when both Israel and the Gaza Strip's Islamist rulers announced unilateral cease-fires on January 18.
Israel, which controls all border crossings except for Rafah (which is managed by Egypt), has kept Gaza closed to all but essential supplies since June 2007, when Hamas seized power after elections and ousted forces loyal to Palestinian president Mahmud Abbas.
Abu Marzouk said his Islamic group is seeking clarification on an Israeli offer to allow a partial opening of the crossings to some 80 percent.
Arab diplomats in the Syrian capital say that the deal would include releasing 1,000 Palestinians in exchange for Shalit who has been in captivity for more than two-and-a-half years.
"The delegation is going back to Cairo tomorrow to pass our replies to our Egyptian brothers and get the Israeli answers on certain demands," Abu Marzouk said.
"As the Israelis are keen on freeing Shalit, we are also determined that they carry out our demands and release the women, children and other brothers in Israeli jails," he added, in reply to a question.
Abu Marzouk refused to discuss details but said that if the Israelis were serious, "I think we will witness progress soon."
By CBS News' George Baghdadi reporting from Damascus.