Israeli war cabinet member Benny Gantz has warned that unless Hamas frees all hostages held in Gaza by 10 March an offensive will be launched in Rafah, reports BBC.
It is the first time Israel has said when its troops might enter Gaza's overcrowded southern city.
Global opposition is growing to such an attack in Rafah, where some 1.5 million Palestinians are sheltering.
Earlier, the UN public health agency said a key Gaza hospital had ceased to function following an Israeli raid.
The World Health Organization (WHO) said it had not been allowed to enter Nasser hospital in Khan Yunis, north of Rafah, to assess the situation.
The Israel Defence Forces (IDF) entered the complex on Thursday, saying intelligence indicated hostages taken by Hamas were being held there.
The IDF has described its operation in Nasser as "precise and limited", accusing Hamas of "cynically using hospitals for terror".
Speaking on Sunday, Mr Gantz, a former defence minister, said: "The world must know, and Hamas leaders must know - if by Ramadan our hostages are not home, the fighting will continue everywhere, to include the Rafah area".
Ramadan - the Islamic holy month of fasting - this year begins on 10 March.
Mr Gantz added that Israel would act in "a co-ordinated manner, facilitating the evacuation of civilians in dialogue with our American and Egyptian partners to minimise civilian casualties".
The Israeli war cabinet consists of the country's top security officials. It was formed several days after Hamas-led gunmen attacked Israel on 7 October, killing at least 1,200 people and taking 253 hostages. Hamas is still holding about 130 hostages in Gaza, Israel believes.
Mr Gantz's reference to Egypt may serve to heighten speculation that Israel expects some Palestinians to cross out of the Gaza Strip and seek shelter on the Egyptian side of the border, where the authorities appear to be building a large walled enclosure for this purpose, says the BBC's diplomatic correspondent Paul Adams.
But Israeli officials have yet to give any details of an evacuation plan, he adds.
With exactly three weeks to go before the start of Ramadan, reports from Rafah say that a few people are leaving, heading west towards the coast, but that most are still waiting, unsure what to do.
Despite international pressure, Israeli Prime Minister Benjamin Netanyahu has vowed to launch a ground assault on Rafah to eliminate Hamas gunmen there.
Egypt and some other Arab countries have repeatedly warned that an Israeli offensive in Rafah would risk pushing many Palestinians into Egypt - which they consider unacceptable. Saudi Arabia has vowed "very serious repercussions" if Rafah is stormed.
Internationally there have been many calls for Israel to refrain from storming Rafah, where Palestinians are living in dire conditions. Israel's offensive against Hamas since 7 October has reduced much of the Gaza Strip to ruins.
More than 28,400 Palestinians, mostly women and children, have been killed and more than 68,000 wounded since the war began, according to the Hamas-run health ministry.
The ministry says at least 127 Palestinians have been killed and 205 others injured in the past 24 hours.
Despite the continued fighting in Gaza, efforts to broker a ceasefire between Israel and Hamas have been taking place in Cairo in recent days - although Qatari mediators said progress was "not very promising".
Mr Netanyahu said he sent negotiators following a request from US President Joe Biden, but added they did not return for further discussions because Hamas's demands were "delusional".
Hamas has blamed Israel for a lack of progress towards achieving a ceasefire deal.