The government accuses Hezbollah of being loyal to Syria and Iran
This would give Hezbollah and its allies a veto over key decisions.
The Hezbollah leader, Hassan Nasrallah, blamed the US for obstructing a solution to Lebanon's political crisis by opposing such a move.
The western-backed Lebanese government has repeatedly rejected the opposition's demand for powers of veto.
The government has proposed reforming the cabinet to give the president a casting vote.
Hezbollah and its allies have been demanding a third of the cabinet seats since the 2006 war with Israel - which Hezbollah regards as a victory - but until now they had not publicly linked the issue to a presidential vote.
Druze leader Walid Jumblatt, who is aligned with the government, said Hezbollah was making impossible demands and was more loyal to Syria and Iran than to Lebanon.
The dispute between the government and opposition has left Lebanon without a president for more than five weeks.
A parliamentary session to elect a president was postponed for the 11th time on 28 December, and is now due to begin on 12 January.
Government and opposition agree on that the next president, traditionally a Maronite Christian and elected by parliament, should be the head of the army, Gen Michel Suleiman.
But they disagree over the shape of a future government.
The wider political crisis has paralysed the government and parliament for more than a year, and spilled over into armed clashes and political assassinations.
Sheikh Nasrallah's comments came in an interview with a private Lebanese TV station, and were aired simultaneously by Hezbollah's al-Manar TV.
"A solution lies in a partnership through a constitutional guarantee (and) through a veto power for the opposition, which represents more than half of the Lebanese people," he said.
"As long as there is a US decision not to give the opposition a veto power, this means there won't a presidential election," he said.
"[The government] wants to fully control authority and rejects partnership with the other party," Sheikh Nasrallah said.