Mr Talabani, a rare unifying figure who is seen to rise above the country's ethnic and sectarian fault lines, has been actively involved in trying to mediate the crisis between Iraq's central government and the country's Kurdish minority.
Prime minister Nouri al-Maliki is at the hospital where the president is being treated.
Doctors have not decided whether Mr Talabani will continue to be treated in Baghdad or will be flown to another country for treatment.
An Iraqi cabinet official said Mr Talabani collapsed on Monday and has not regained consciousness.
The Iraqi presidency is seen as a largely ceremonial post, although it does retain some powers under Iraq's constitution. The president must sign laws approved by parliament and has the power to block executions.
Mr Talabani, a member of Iraq's Kurdish minority, has frequently used the post to mediate disputes within the government and among Iraq's various sects and ethnic groups.
He has recently been working to resolve a stand-off between the central government and the Kurds, who have their own fighting force.
The two sides last month moved additional troops into disputed areas along the Kurds' self-rule northern region, prompting fears that fighting could break out.
Mr Talabani last week brokered a deal that calls on both sides to eventually withdraw troops from the contested areas.