"We remind the president that we have open options to deal with the ongoing crisis, but we prefer the national option, which is in harmony with our national unity," Hamas said on Thursday.
Aljazeera said Hamas also accused Fatah of imposing conditions that fell in line with US demands, adding that these had been rejected nationally.
The Islamist group called for a resumption of efforts for forming a national unity government, sidestepping US pressure and working towards advancing Palestinian interests.
Mahir Miqdad, a spokesman for Fatah, rejected the accusation that Abbas and Fatah-linked organisations were impeding the formation of a national unity government.
In an interview with Aljazeera, Miqdad said Fatah was ready to consider all points of view that might help reach agreement on the subject of a national unity government.
In other developments, Qatar's first deputy prime minister and foreign minister held talks on the Palestinian situation with Bashar al-Assad, the Syrian president, and Hamas's exiled political leader Khaled Mashaal in Damascus on Thursday.
A Hamas official told The Associated Press that Sheikh Hamad bin Jassem bin Jabor al-Thani came to Syria to try to end the standoff between the ruling Hamas group and the Fatah Party in the Palestinian territories.
The official, who asked to remain anonymous because of the sensitivity of the talks, said Qatar sought to "bring the point of view closer between Hamas and the Palestinian President Mahmoud Abbas" and remove tension between the two sides.
Conditions in the West Bank and Gaza have rapidly deteriorated since Hamas defeated Abbas's Fatah party in January elections and Western countries cut off funds to the Palestinians.
Violence erupted on Sunday when Hamas fighters confronted members of the Fatah-dominated security forces who were protesting at the government's inability to pay their wages.
Ten people were killed and around 100 wounded in ensuing street battles over three days.
Syria's official news agency Sana said on Thursday that Sheikh Hamad gave al-Assad a letter from the Amir of Qatar on "the latest developments in the region and bilateral relations between the two countries".
It said the two sides discussed during the meeting "the latest on inter-Palestinian dialogue and the need to support Palestinian national unity at this sensitive time in the region".