The centre-right opposition refused to back the incoming government's preferred candidate, a former communist.
No candidate obtained the required two-thirds majority in a fresh vote in Tuesday's second round and Silvio Berlusconi, the outgoing prime minister, said there was currently no room for agreement with the centre-left.
Barring any last-minute deal, the third ballot on Tuesday night was also almost certain to be inconclusive.
The developments mean Romano Prodi's centre left will have to decide whether to push its candidate through, starting with Wednesday's fourth ballot - when the majority needed to elect is greatly reduced - or continue to seek a compromise.
The parliamentary election for Italy's 11th postwar head of state is the first test for Prodi since he narrowly won last month's general election.
It has underscored the difficulty he will have in pushing through his agenda with a slim majority.
The post of president is largely ceremonial but under the constitution the head of state names the prime minister and dissolves parliament - prerogatives which could be crucial for Prodi as he prepares to form a government.
Prodi cannot take office until the new president gives him a mandate.
Silvio Berlusconi casts his ballot
The centre left wants the post to be given to Giorgio Napolitano, an 80-year-old senator-for-life of the Democrats of the Left, Italy's former communist party.
The two-thirds majority needed to elect the president in the first three ballots means a successful vote is impossible unless the two blocs agree on a compromise candidate.
In the first of Tuesday's two votes, most ballots were left blank, as negotiations continued behind the scenes.