President Hamid Karzai and his US counterpart Barack Obama held a telephonic conversation on Sunday night, his office said in a statement.
Both the presidents held talks after Obama landed at the Bagram Airbase in central Parwan province following a secret trip to Afghanistan on Sunday evening.
A statement from the presidential palace said Obama congratulated Karzai on holding successful presidential and provincial council elections and the key role played by Afghans in the democratic process.
The US president also expressed optimism the second round of ballot would be held peacefully. Also, the Afghanistan peace process came under discussion between the two leaders.
The statement quoted Karzai as saying, the oath-taking ceremony for the new Afghan president would take place on August 2, hoping a high-ranking US officials would attend the ceremony.