The White House said Tuesday it was hopeful "Israel" and the Palestinian Resistance group Hamas could "close the remaining gaps" in their ceasefire talks.

"A close assessment of the two sides' positions suggests that they should be able to close the remaining gaps, and we're going to do everything we can to support that process," National Security Council Spokesperson John Kirby told reporters.

Kirby said the fact that "Israel", Hamas' mediator Qatar, and CIA chief William Burns were all going to be present at the talks in Cairo indicated the talks were at an advanced stage.

"Everybody's coming to the table," the Spokesperson said. "That's not insignificant."

Asked what the assessment was based on, Kirby mentioned that "there were amendments offered" by Hamas and "it's based on our understanding of where the text is right now."

He expressed the White House's hope for news "very very soon" but said it would be "foolhardy" to predict when the negotiations might bear fruit.

Touching on the Israeli ground invasion of the densely-populated city of Rafah, the spokesperson said the White House has been told by Israeli officials that the attack on Rafah in southern Gaza last night was "limited and designed to cut off Hamas' ability to smuggle weapons" into the besieged Strip.

The Israeli operation was of "limited scope, scale, and duration," he added.

In a ground invasion, Israeli occupation forces occupied the Rafah border crossing with Egypt, despite Hamas' earlier announcement that it had accepted a ceasefire proposal presented by mediators Egypt and Qatar.

Kirby mentioned that the United States will be monitoring the Israeli military actions in Rafah closely, adding that his country wants to see the Rafah and Karem Abu Salem crossings -- the only routes to deliver aid to southern Gaza -- restored as soon as possible.

Read more: Israeli officials believe they 'got played' by US on Gaza deal: Axios

'Israel' would invade all of Gaza if captives were not retrieved: Gallant 

On the other side, Israeli Security Minister Yoav Gallant made it clear on Tuesday that "Israel" was prepared to "deepen" its ground invasion of Gaza if negotiations fail to secure the release of captives held by the Palestinian Resistance in the besieged Strip.

Touring the Rafah area after Israeli forces occupied the Rafah border crossing, Gallant claimed "Israel" is prepared to "make compromises" to retrieve the captives but warned that "if that option is removed, we will go on and deepen the operation."

He also mentioned that the invasion of Rafah "will continue until we eliminate Hamas in the Rafah area and the entire Gaza Strip," or until the first captive is retrieved.

According to the Israeli minister, if the captives are not retrieved, "Israel" would expand its invasion "all over the Strip, in the south, in the centre and in the north."

His comments came after Israeli negotiators arrived in Cairo for the latest talks toward a prisoner exchange and ceasefire deal.

Israeli Prime Minister Benjamin Netanyahu claimed Tuesday that the occupation of the Rafah crossing is "an important step on the way to destroying the remaining military capabilities of Hamas."

Read more: Rafah may be 'sea-change moment' for Democrats with 'Israel': Axios