He said there was no dialogue unless it recognises the supremacy of the Palestine Liberation Organisation (PLO).
He was responding to a proposal by the Islamist movement Hamas for the Palestinians to replace the PLO, which is dominated by Abbas and the factions loyal to him.
"Now we say ... no dialogue with those who reject the Palestine Liberation Organisation," Abbas said in a speech at a Palestinian hospital in the Egyptian capital Cairo.
"They must admit without equivocation or ambiguity that the organisation is the sole and only representative of the Palestinian people. Then there will be dialogue," he added.
The PLO has represented the Palestinians since 1964 but the more recently created Islamist movements, such as Hamas and Islamic Jihad, have never been part of it.
The Islamists reject most of the agreements the PLO has made with Israel, including the 1993 Oslo accord.
Meshaal has advocated a new umbrella body to represent Palestinians in Gaza, the West Bank and in the diaspora.
His proposal was echoed on Friday in similar statements to cheering crowds by a senior Hamas political leader, Khalil al-Hayya.
Hayya said the PLO was dead, sent to the morgue by those who founded it. "It is high time the Palestinian people have a new leadership," he added.
ReutersLast Mod: 02 Şubat 2009, 12:00