"On the assumption that the government tonight approves the framework for the release of the hostages, then a 24-hour period comes in where the supreme court hears the petitions, on the assumption that it rejects the petitions and allows the government to go ahead - which I think is a reasonable assumption - that means we can start moving on this deal on Thursday, hopefully seeing people come home immediately," Regev told UK media.
He added that the vote is still pending but may happen within an hour or two.
Regev further said Israel could consider extending the ceasefire with Hamas under the deal if more hostages were released.
"Theoretically, if we were to get more people out we would consider extending the ceasefire," he said.
US State Department spokesman Matthew Miller said Tuesday that the Israeli cabinet was meeting earlier in the day to discuss the deal to release hostages held by Hamas in Gaza.
Netanyahu said Tuesday that progress had been made in the negotiations on the release of hostages from the Gaza Strip, expressing hope that soon there will be "good news" on the matter.
Earlier in the day, the head of the political bureau of Hamas, Ismail Haniyeh, said that a ceasefire agreement was close to being concluded as the movement handed over its response to Qatar and other mediators in negotiations with Israel. After that, Israeli media reported, citing an unnamed high-ranking official, that the deal would include the release of dozens of hostages in exchange for a temporary ceasefire.
Qatari Foreign Minister spokesman Majed Mohammed Al-Ansari said Tuesday that negotiations between Israel and Hamas on a truce have reached the final stage, but details of the agreement have yet to be announced.