Israel gave Hamas a week to agree to a cease-fire deal, or the invasion of Rafah will begin, Egyptian officials told the Wall Street Journal on Friday.
Egyptian officials claimed that Hamas is seeking a long-term truce and guarantees from the US that a cease-fire will be respected by Israel.
Hamas expressed concern that the latest proposal is still too vague and gives Israel room to restart the fighting.
Yahya Sinwar, Hamas's Gaza chief, has yet to make his view known, further complicating the situation.
Israeli figures from across the right slammed the deal when it was announced earlier in the week.
Proposal details
The proposal calls for an initial period of up to 40 days of cessation of violence, during which Hamas would release up to 33 hostages, with the possible negotiation of a long-term cease-fire to develop.
Next, there would be a cease-fire of at least six weeks, during which Hamas and Israel would agree on a larger hostage release and an extended pause in fighting that could last up to a year.
Egypt invited senior Hamas officials to Cairo to continue negotiations, and Hamas confirmed they would be present.
Neither the US nor Israel commented on the report.
Netanyahu has persisted in insisting that Israel will enter Rafah regardless of whether a deal is made or not.
Hamas was concerned that Netanyahu was attempting to provoke them into refusing the proposal, to provide justification for invading Rafah, and thus blame Hamas for the failure of negotiations.
According to Egyptian officials, Hamas is expected to respond to the proposal with a counteroffer rather than outright rejection.
The US told Hamas that they should accept the deal as is and not push for further concessions.