JERUSALEM — European nations are offering to help enforce the cease-fire in the Gaza Strip, a scenario that could provide key international backing for maintaining the peace and step up the pressure on Hamas militants to relinquish power.

The European plan remains vague, and it is unclear whether Israel or the Palestinians will agree.

But a European presence in Gaza could go a long way toward meeting two key demands: the Palestinians’ insistence on freer movement in and out of the territory, and the Israeli requirement that Hamas be kept in check.

French President Francois Hollande laid out the case for European involvement on Thursday, telling international diplomats that Europe could help oversee the destruction of tunnels used by Hamas militants and monitor the territory’s border crossings with Israel and Egypt.

“It is necessary to move toward an end to the blockade and a demilitarization of the territory,” he said, indicating that international supervision could help pave the way for a return of Hamas’ rival, the Palestinian Authority, to Gaza.

President Mahmoud Abbas, who heads the authority, is eager to regain a foothold in Gaza, seven years after Hamas violently overran the territory. With the international community shunning Hamas as a terrorist group, Abbas would be likely to operate Gaza’s borders and oversee internationally funded reconstruction efforts.

The French proposal, Hollande said, would “finally give the Palestinian Authority the means to respond to the humanitarian crisis and to begin reconstruction.”

Earlier this month, while the fighting was still raging, the European Union expressed its readiness to contribute to a “sustainable solution” for Gaza, offering help in postwar reconstruction and monitoring efforts.

Among other things, it mentioned the possible revival of an operation that helped operate Gaza’s volatile border crossing with Egypt.

Israel and Hamas militants battled for 50 days this summer before reaching a truce on Tuesday. More than 2,100 Palestinians were killed, including hundreds of civilians. Seventy people on the Israeli side, including six civilians, died. An estimated 100,000 Gazans have been left homeless, and reconstruction is expected to take years.

The cease-fire brought an immediate end to the warfare — the third round of heavy fighting between the two sides since Hamas came to power in 2007 — but left key matters unresolved.

While Israel agreed to loosen a longstanding blockade to allow humanitarian aid and reconstruction materials into Gaza, many of the border restrictions will remain in place. Hamas, meanwhile, rejected Israel’s demands that it disarm.

These deeper issues are to be addressed in indirect talks in Egypt next month.

Hollande’s comments also reflected a European desire to take a bigger role in Mideast diplomacy, an area traditionally dominated by the U.S.

“It is Europe that does so much to rebuild, develop Palestine. It is Europe that also must pressure each one and not just be an echo chamber for complaints,” he said.

Whether the sides are ready to accept external involvement is unclear.

Paul Hirschson, a spokesman for Israel’s Foreign Ministry, said that the country is ready to consider any proposal but must be convinced that monitoring will work.

Israel’s main concern, he said, is to prevent Hamas from smuggling more weapons through reopened border crossings. Europe’s border-monitoring operation, EUBAM, halted its operations after Hamas overran Gaza in 2007.

“We’re not saying no to anything. But it has to be something that’s workable,” Hirschson said.

Complicating matters is that the European Union, like Israel and the U.S., considers Hamas a terrorist group and has no direct contact with the organization.

At the minimum, an international presence in Gaza would require Hamas to allow Abbas’ forces to help operate the border crossings — something it has expressed a willingness to do.

But it would also step up pressure on Hamas to give up its military capabilities, such as the network of tunnels used to smuggle weapons and stage attacks on Israel.

In Gaza, Hamad al-Rakeb, a Hamas spokesman, described Hollande’s proposal as “mixing poison in the honey.”

Riyad Mansour, Abbas’ ambassador to the United Nations, on Thursday welcomed the idea of international monitors as a “useful deterrent” to more fighting. He acknowledged, however, that disarming Hamas is “not realistic.”

Israeli Finance Minister Yair Lapid said Thursday that international aid must be conditioned on Hamas disarming. During the war, Hamas fired thousands of rockets and mortars into Israel.

“They want to rehabilitate Gaza, they need to understand that they will have to disarm. Residents of the south cannot continue to live this way,” Lapid said.

Donor nations are scheduled to meet in October for a conference on funding reconstruction of Gaza, with the expectation that hundreds of millions of dollars in aid will be channeled through Abbas.

Yossi Beilin, a former Israeli deputy foreign minister, said Europe’s readiness to help could prove vital in reopening Gaza’s borders and returning Abbas’ government to power there. He said while demilitarization would be difficult, it would be impossible without international pressure.

“If all the parties support it, and there is no reason for Israel, Egypt and the Palestinian Authority to reject it, how can Hamas really reject it? Only by violence,” he said. “It’s difficult to understand them, but it is unlikely that they will turn against the Europeans now.”