The two sides have made several attempts at reconciliation.
The group said in an emailed statement that it had agreed to dismantle the administrative committee that runs Gaza and hold nationwide elections.
The group said it was "responding to the generous Egyptian efforts, which reflect the Egyptian desire to end the split and achieve reconciliation, and based on our desire to achieve national unity".
Some polls show that if parliamentary elections were held now, Hamas would win them in both Gaza and the Israeli-occupied West Bank, the seat of Abbas's Palestinian Authority.
"Hamas dissolves the administrative committee and invites the consensus government of Prime Minister Rami Hamdallah to come to the Gaza Strip immediately to practice its missions and carry out its duties", said Hamas statement issued from Cairo on Sunday predawn.
Paving the way for possible reconciliation between the West Bank and Gaza, Hamas said Sunday it would dissolve the Gaza administrative committee, allowing a unity government to work in its place and agree on general elections.
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Fatah Vice Chairman Mahmoud al-Aloul welcomed Hamas's announcement, but said his party would first like to confirm its veracity.
Hamas would like Abbas to lift the various punitive measures he has taken against Hamas in Gaza over the past four to five months including cuts to budgets allocated to Gaza for electricity, medical services, salaries and other purposes.
Ismail Haniya, who heads Hamas, travelled to Cairo last week for reconciliation discussions, and Abbas sent a delegation of representatives to Egypt as well. However, in July 2014, Israel waged a large-scale military offensive on the Gaza Strip that last 50 days.
It added that it had decided to make the announcement in response to Egyptian efforts to achieve reconciliation and end internal Palestinian divisions.
"This position puts President Mahmoud Abbas and his Fatah movement in front of the Egyptian efforts and in front of the Palestinian people before a real test in order to achieve our people's aspirations, achieve a real unity and a real partnership", said Barhoum.
Palestinian president Mahmoud Abbas's western-backed Palestine Authority, which is controlled by Fatah, fought a war with Hamas over Gaza in 2007, which led to Hamas taking over.
Officials also said more information was needed before considering it a done deal.