The statement comes after Hamas leaders held talks with Egyptian officials last week, and with the Gaza Strip run by the Palestinian Islamist movement facing a mounting humanitarian crisis.
Hamas said it had agreed to key demands made by Fatah: dissolving the so-called "administrative committee" created in March, while saying it was ready for elections and negotiations toward a unity government.
It called on the Palestinian Authority government based in the Israeli-occupied West Bank "to come to Gaza to exercise its functions and carry out its duties immediately".
It was unclear, however, whether the steps would result in further concrete action toward ending the deep division with Fatah.
Also Read
Hamas for now continues to run a de facto separate administration in the Gaza Strip and is in charge of the security forces there.
Previous attempts to resolve the split have repeatedly failed.
Abbas's Fatah welcomed the announcement, saying it followed "extensive meetings" between its own representatives and Egyptian intelligence officials.
Fatah official Azzam al-Ahmad said a bilateral meeting with Hamas would be organised to begin working out a way forward.
The Gaza Strip has been under an Israeli blockade for around a decade, while its border with Egypt has also remained largely closed in recent years.
Abbas is currently in New York for the UN General Assembly meeting, where he is set to speak on Wednesday.
"It is critical that the grave humanitarian situation in Gaza, most notably the crippling electricity crisis, be addressed as a priority."
In Sunday's statement, Hamas spoke of the "dissolution" of the administrative committee, which was seen as a rival government to Abbas's administration.
Disclaimer: No Business Standard Journalist was involved in creation of this content