In an interview to be shown on ITV1's Jonathan Dimbleby politics programme on Sunday, Abbas accepted there were difficulties posed by Hamas' refusal to acknowledge Israel, but held back from saying directly he would quit if they continued.
"We could reach a point where I cannot perform my duty, then I will not continue sitting in this place, against and in spite of my convictions," he said, speaking in Arabic and dubbed into English.
"If I can do something then I will continue, otherwise I won't. From the beginning, I said ... that if I fail I will resign," he added.
Hamas's landslide victory over Abbas's Fatah faction in the 25 January Palestinian election paved the way for the group to form a new cabinet and knocked down hopes Middle East peacemaking might be revived.
Hamas, sworn to Israel's destruction, has rejected talks with Israel as a waste of time.
The group has masterminded nearly 60 suicide bombings against Israelis since a Palestinian uprising began in 2000, but has largely adhered to a truce declared last March.
"If I can do something then I will continue, otherwise I won't. From the beginning, I said ... that if I fail I will resign"
George Bush, the US president, has urged the international community to make clear to Hamas that it must recognise Israel's right to exist or else the Palestinian Authority which it would lead would be denied direct aid.
Hamas has so far not been swayed, saying Western threats to cut off aid amount to blackmail and alternative sources of funding can be found.
Abbas described the withholding of aid as creating an economic crisis.
"Hamas has to say that it accepts all the commitments made by the Palestinian Authority," Abbas said.
"When it says that ... it will be sufficient for Hamas to become accepted and Hamas then will not give the pretext to anyone to say that this is a terrorist government or this government wants to (overthrow) Israel," he added.
However, he said that the swearing in of a Hamas government would not in itself trigger a crisis.
"When Hamas is sworn in, talks and discussions will have to start, as well as the dialogue with the Israelis in different ways," he said.