Prosecutors say Mr Bashir's government sought to wipe out three ethnic groups.
They argued last year that it had intended to destroy the Fur, Masalit and Zaghawa peoples.
If genocide charges are now brought, they will be the first to be issued by the ICC against a sitting head of state.
African and Arab leaders have rallied around Mr Bashir and several nations have refused to honour the existing warrant.
Mr Bashir was travelling to Qatar for a one-day visit on Wednesday for talks on peace in Darfur with the Qatari leader, Sheikh Hamad bin Khalifah al-Thani, Sudanese state radio said.
The UN says 300,000 people have died in the Darfur conflict since 2003.
'Error of law'
"The pre-trial chamber is directed to decide anew," presiding judge Erkki Kourula said, upholding the appeal lodged by prosecutor Luis Moreno-Ocampo.
James Copnall, BBC News, Khartoum
The ICC's appeal court ruling reopens a painful wound for President Omar al-Bashir and his supporters. Genocide is a word with the power to infuriate the president and provokes his supporters into tirades whenever US officials pronounce it.
But an ICC charge against President Bashir has the potential to stir up much more than just heated words. When an arrest warrant was issued against him last March, 13 foreign NGOs were kicked out of northern Sudan.
Khartoum probably cannot afford a further weakening of its relationship with the international community right now and everyone is still waiting to see what the pre-trial chamber will decide but the anger provoked by the renewed possibility of a genocide charge is real, and it is difficult to predict its course.
Judge Kourola said the standard of evidence set in the previous ruling had been "higher and more demanding than what is required".
The prosecution had argued that the time to provide definitive levels of proof was during the main trial and not when they were just seeking to bring charges.
"The decision by the pre-trial chamber not to issue a warrant in the respect of the charge of genocide was materially affected by an error of law," Judge Kourola said in court.
He stressed that the appeals court was not itself ruling that Mr Bashir was criminally responsible for the crime of genocide.
Reacting to the ruling in Khartoum, Rabie Abdelati, a senior information ministry official, accused the court of seeking to obstruct democratic elections due to be held in April.
"This procedure of the ICC is only to stop the efforts of the Sudanese government towards elections and a peaceful exchange of power," the official said.
International Criminal Court judges reverse the ruling
"Our president has visited a lot of countries, and right now he is in Qatar for the second or third time. We are not bothering actually what the ICC will say, whether it includes genocide or not."
A spokesmen for one of Darfur's most powerful rebel groups, the Justice and Equality Movement (Jem), welcomed the ruling in The Hague, with one saying the Jem might reconsider taking part in peace talks.
"This is a correct decision," Ahmed Tugud told Reuters news agency by telephone from the Qatari capital, Doha.
"We believe that what we have seen on the ground in Darfur amounts to a crime of genocide. Now we are assessing our situation on whether it is ethically possible to negotiate with a government accused of committing genocidal crimes against our people."
Mr Bashir has avoided arrest thanks to support from other leaders. Since the warrant was issued he has visited Qatar, Ethiopia and Zimbabwe among other nations, and the African Union has consistently supported him.
The AU's most senior diplomat, Jean Ping, hit out at the ICC in the run-up to its decision, accusing the court of targeting African nations.
"We are not for a justice with two speeds, a double standard justice - one for the poor, one for the rich," he said.
Mr Bashir's government is accused of backing Arab militias who killed thousands of black African Darfuris.
Mr Bashir has repeatedly said he had no control over the actions of people on the ground in Darfur at the height of the violence in 2003 and 2004.
This page is best viewed in an up-to-date web browser with style sheets (CSS) enabled. While you will be able to view the content of this page in your current browser, you will not be able to get the full visual experience. Please consider upgrading your browser software or enabling style sheets (CSS) if you are able to do so.