Aliyev, 80, who has a history of heart problems and has not appeared in public since he was hospitalized on July 8, left the Gulhane military hospital in Ankara on Wednesday to travel to the Cleveland Clinic in Ohio, the embassy said in a statement.
The move comes amid increasing concern about the president's health and signs that his son, Ilham Aliyev, was preparing to take over the country's leadership.
The embassy gave no details about the president's condition.
Aliyev was taken from the hospital in an ambulance and was traveling on an ambulance plane to the United States, Turkey's Anatolia news agency reported.
Reporters at Ankara's Esenboga Airport could not see Aliyev board the Russian Emergency Situations Ministry Il-62 jet. Authorities had placed a bus in front of the staircase leading into the plane, apparently to block Aliyev from view.
On Monday, Azerbaijan's parliament approved what lawmakers said was Aliyev's nomination of Ilham Aliyev as prime minister. According to constitutional changes approved last year, the prime minister becomes acting president if the president is incapacitated or resigns. Azeri opposition politicians have disputed the appointment, claiming the president is already dead.
There have been conflicting reports about the president's health.
Azerbaijan's ambassador to Turkey, Mamed Aliyev, has said the president was being treated for heart problems. He told Itar-Tass on Wednesday that Aliyev's condition was stable.
CNN-Turk private television, citing hospital officials, reported early Wednesday that Aliyev was connected to a respirator, could not talk, and was only occasionally conscious. Turkey's private NTV news channel reported that Aliyev needed a heart and kidney transplant.
Doctors have said that Aliyev's condition is serious and warned him against flying home, saying he was too frail.
A presidential election is scheduled for October, but it is not clear whether Aliyev will remain a candidate.
The Azeri Cabinet on Wednesday agreed to allow Ilham Aliyev to go on an unpaid vacation so that he could take part in the presidential campaign, the state information agency Azertaj said. Elections are scheduled for Oct. 15, and both Aliyevs are on the ballot.
First Deputy Prime Minister Artur Rasizade will carry out Ilham Aliyev's duties as acting prime minister.
A Message from The Moscow Times:
Dear readers,
We are facing unprecedented challenges. Russia's Prosecutor General's Office has designated The Moscow Times as an "undesirable" organization, criminalizing our work and putting our staff at risk of prosecution. This follows our earlier unjust labeling as a "foreign agent."
These actions are direct attempts to silence independent journalism in Russia. The authorities claim our work "discredits the decisions of the Russian leadership." We see things differently: we strive to provide accurate, unbiased reporting on Russia.
We, the journalists of The Moscow Times, refuse to be silenced. But to continue our work, we need your help.
Your support, no matter how small, makes a world of difference. If you can, please support us monthly starting from just $2. It's quick to set up, and every contribution makes a significant impact.
By supporting The Moscow Times, you're defending open, independent journalism in the face of repression. Thank you for standing with us.
Remind me later.