Details of the agreement have not yet been released and a press conference was being planned for Thursday afternoon in the Egyptian capital, where talks have been taking place since Tuesday.
Mr Abbas welcomed the deal in comments to AFP and said he considered it a "final agreement to end the division" - though many details remain to be resolved and previous reconciliation attempts have repeatedly failed.
Hamas leader Ismail Haniya's office said in a statement, without giving further details, that "an agreement was reached today between Hamas and Fatah under Egyptian sponsorship."
An official from Abbas's Fatah movement said the Palestinian president was now planning to travel to the Gaza Strip within a month as part of the unity bid in what would be his first visit in a decade.
Sanctions taken by Abbas against Hamas-controlled Gaza will also soon be lifted, the Fatah official said.
The deal includes 3,000 members of the West Bank-based Palestinian Authority's police force redeploying to Gaza, a member of the negotiating team told AFP on condition of anonymity.
The figure is however a fraction of the more than 20,000 police officers employed separately by Hamas.
Another party to the negotiations, speaking on condition of anonymity, said the agreement would see Palestinian Authority forces take control of the Rafah border crossing between Gaza and Egypt.
He added that all Palestinian factions would begin wider negotiations on the formation of a unity government in the coming two weeks.
One of the key issues has been punitive measures taken by Abbas against Gaza in recent months, including reducing electricity payments that left the territory's residents with only a few hours of power a day.
"All the measures taken recently will end very shortly," Zakaria al-Agha, a senior Fatah leader in the Gaza Strip, told AFP.
The two sides had been meeting in the Egyptian capital this week with the aim of ending the crippling decade-old split between the rival factions.
Hamas seized Gaza from Fatah in a near civil war in 2007 and the two factions have been at loggerheads ever since. Multiple previous reconciliation efforts have failed.
Egypt has been keen to improve security in the Sinai Peninsula which borders Gaza and where jihadist rebels have fought a long-running insurgency.
An Egyptian source close to the talks said intelligence chief Khaled Fawzi had followed the talks closely.
Last month, Hamas agreed to cede civil power in Gaza to the Palestinian Authority but the fate of its vast military wing remains a significant issue for the two sides.
Islamist movement Hamas is blacklisted as a terrorist organisation by the United States and the European Union.
It has fought three wars with Israel since 2008 and the blockaded Gaza Strip has seen deteriorating humanitarian conditions.
Faced with increasing isolation and a severe electricity shortage, Hamas has reached out to Egypt for help, hoping to have the Rafah border opened.
Egypt has also agreed to provide fuel to the Gaza Strip for electricity generation.
In return, Cairo pressed Hamas to move forward on reconciliation with Fatah.
Previous attempts at reconciliation have repeatedly failed, and many analysts are treating the latest bid with caution, waiting to see if actual change will occur on the ground.
Last week, Palestinian Authority prime minister Rami Hamdallah visited Gaza for the first time since 2015 and his ministers took formal control of government departments in the territory.
But the move was seen as mainly symbolic, with Hamas still effectively in charge in the Palestinian enclave of two million people bordered by Egypt, Israel and the Mediterranean Sea.
One of the key sticking points will be the fate of Hamas's 25,000-strong military wing, the Ezzedine al-Qassam Brigades.
Reconciliation could also pose a dilemma for international efforts to reach an Israeli-Palestinian peace deal since Hamas has not recognised Israel, unlike the Abbas-led Palestine Liberation Organisation.