Israel wants Abbas to do more to rein in on Hamas fighters and has been considering providing his forces with additional arms and ammunition.
Before Hamas took full control of the Gaza Strip almost two weeks ago, Abbas had intended to send the Badr Brigade there in an effort to stop armed groups from firing makeshift rockets into Israel.
The proposed deployment was never initiated.
"The refusal by the US, EU et al to deal with Hamas reflects their disrespect for the wishes of the Palestinian people"
Elise, Bemidji, USSend us your views
If you are in the Gaza Strip, you can also mail your opinions to:
Some accounts have suggested that the Badr Brigade has less than 1,000 fighters with various levels of training but analysts say Badr is Fatah's best-trained and best-equipped fighting force, aside from Abbas's presidential guard.
Western officials have visited the main Badr training base in Jordan but a formal evaluation of their strength has never been made.
Badr is considered to be more loyal to Fatah than other forces. It also has strong ties to the Jordanian king. It would reinforce his presidential guard and other Fatah-dominated security services.
US Lieutenant-General Keith Dayton, the US security coordinator between Israel and the Palestinians, has yet to publicly take a position on the potential deployment.
The United States has so far not provided Badr troops with training or equipment, but has assisted with coordinating preparations for their deployment, sources said.