Neither the governing coalition, which is opposed to Syrian influence on the country, nor the Hezbollah-led opposition have enough seats in parliament to secure a two-thirds quorum for the election, delayed repeatedly since September 25.
The presidency has been vacant since pro-Syrian Emile Lahoud's term ended four weeks ago.
An article in the constitution bans senior public servants from running for office.
The opposition says, however, no constitutional amendment is necessary. It also considers the government and any decisions it takes illegitimate since its own ministers resigned last year.
Ali Hassan Khalil, an adviser to opposition leader Nabih Berri, the parliament speaker, said of the draft amendment: "This escalation indicates the intention to obstruct a deal and the closing of doors for any initiatives."
Theoretically the draft law passes to parliament for approval. But Khalil said Berri would not receive the draft law because he considers the government unconstitutional.
Berri was quoted in Ad-Diyar newspaper as saying that electing Suleiman did not require a constitutional amendment and that he would continue to call parliamentary sessions to try to elect a president.
Ghazi Aridi, the information minister, said after reading out the government's decision: "The government considers itself legitimate and constitutional and cannot give up its responsibilities and the performance of its duties."
Berri last week postponed the parliamentary election vote for the tenth time, to December 29.
The crisis is viewed as Lebanon's most serious internal political confrontation since the 1975-1990 civil war.