The Turkish and Armenian teams will face each other for the first time in a World Cup qualifier on Saturday.
The two countries have no official ties and their shared border remains closed.
Turkey has rejected Armenia's campaign for the killings of some 1.5m of its citizens, by Ottoman Turks between 1915 and 1917, to be classified as genocide.
More than a dozen countries, various international bodies and many Western historians have recognised the killings as genocide.
Turkey admits that many Armenians were killed but it denies any genocide, saying the deaths were a part of World War I.
Climate of friendship
Mr Gul will become the first Turkish head of state to visit the Armenian capital, Yerevan, when he attends the match with his counterpart, Serge Sarkisian.
This visit is likely to be highly controversial in Turkey, says the BBC's Pam O'Toole.
The Armenian invitation has already sparked a major debate, with some nationalists regarding the fact that the president was even considering taking it up as a betrayal of the country's national interests.
Some columnists have questioned why Mr Gul should visit a country they refer to as Turkey's enemy.
The main opposition party has said such a trip would be a major deviation from state policy, while others see it as a valuable chance to break the logjam in relations, our correspondent says.
A statement on President Gul's website expressed hope that the trip would provide an opportunity for the two countries to understand each other better and create a new climate of friendship in the region.
Turkey and Armenia have had no diplomatic ties since Armenia became independent from the Soviet Union in 1991.Their common border has been closed since the war between Armenia and Turkey's ally, Azerbaijan, in the 1990s over the disputed territory of Nagorno-Karabakh.
From BBC News.