"Once a member of the PLO, a movement can always reform it from top to bottom if it holds a majority," he said.
Hamas is not a member of the PLO, which is made up of several Palestinian nationalist groups, with Fatah being the largest.
Gul appealed for reconciliation between Fatah and Hamas, saying it was an unavoidable precondition for the eventual establishment of a Palestinian state.
The Palestinian president also renewed his call for Israel to accept the Arab Peace Initiative set out in 2002 if it wants to resolve the Palestinian-Israeli conflict.
Abbas said that "Israel has no other choice than to accept the Arab peace plan".
He pointed to the initiative, revived in March 2007 at a summit in Riyadh, the capital of Saudi Arabia, as the best way forward in the Middle East.
The initiative would see all Arab nations establish normal relations with Israel in return for an Israeli pull-out from occupied lands and the creation of a Palestinian state with its capital in east Jerusalem.
While citing "positive aspects" in the initiative, Israel never formally accepted it, chiefly because it refers to a right of return for Palestinians made refugees by the 1948 founding of the Jewish state.
Promoted by Saudi Arabia, the initiative was embraced by all Arab League member nations at a summit in Beirut in March 2002.
Abbas said Turkey, which is not a league member, supports it.