Al Jazeera's Zeina Awad, reporting from Tel Aviv, said: "Hamas has said on the record that there has been progress, but they caution against what they see as a media storm over the issue.
"We have spoken to a large number of high-ranking officials within Hamas, and off the record they have said to us that a deal is at its final stages. It's down to numbers, and it's down to lists of prisoners that will be exchanged for Shalit.
"The efforts to win Gilad Shalit's release are continuing and taking place outside the media spotlight"
The office of Binyamin Netanyahu, Israeli prime minister
"We understand from Hamas that the contentious issues are that Hamas wants a comprehensive list that includes Hamas prisoners as well as Fatah prisoners, as well as so-called Israeli-Arabs or Palestinians with Israeli passports, and East Jerusalemites.
"Israel simply does not want this, instead it wants to narrow the list to those from the West Bank and Gaza."
Intermittent negotiations between Israel and Hamas have been conducted indirectly, mainly through Egypt, since Shalit was seized by fighters from Hamas and allied groups in a cross-border raid in 2006.
Germany has also been involved in mediating in recent months.
Al Jazeera's Amr el-Khaky, reporting from Cairo, said that the Egyptians and Germans would put a new deal to the Hamas delegation on Tuesday morning.
"There are still some differences between Hamas and Israel over the release of 70 Hamas members, most of whom were arrested after Shalit was taken hostage," he said.
There have been reports that Israeli officials do not want some of the prisoners to be allowed to return to Palestinian territories and are attempting to negotiate that they be freed into exile.
'Time for action'
Shalit's parents met Hagai Hadas, the Israeli official who has been leading the indirect efforts to secure the soldier's release, at the defence ministry in Tel Aviv on Monday.
"Of course we are hoping and want to see Gilad at home after so many years," Noam Shalit, the father, told reporters.
|Hundreds of Palestinians could be freed from Israeli jails in return for Shalit [AFP]
"To my regret I cannot discuss [this] today, and I don't want to. This is not the time for chatter but for action."
Marwan Bishara, Al Jazeera's senior political analyst, said that a deal "seems closer than before".
"This time around there is at least one sign from this week, the Israelis have put all those hundreds of prisoners in three distinct prisons within Israel. Perhaps they are preparing the way for transferring them all as one," he said.
However, the office of Binyamin Netanyahu, the Isaeli prime minister, refused to confirm whether a deal was likely in the coming days.
"The efforts to win Gilad Shalit's release are continuing and taking place outside the media spotlight. We have no intention of commenting beyond this," it said in a statement.
In October, Israel freed about 20 Palestinian women prisoners in exchange for a video showing Shalit in captivity.
The video showed Shalit alive and moving - the first proof of his wellbeing since he was captured.