The developments were the latest in three weeks of anti-government demonstrations that have engulfed Sudan.
Al-Bashir told a gathering of several thousands of supporters in the capital that he is ready to step down only "through election."
There "are those who conspire against Sudan and seek to attack it. There are no other options but national dialogue and elections," he said in televised comments as supporters chanted "there is no alternative to al-Bashir."
"The decision is the decision of the Sudanese people through the ballot box," added al-Bashir, who then briefly danced on the stage.
Also at the rally, al-Bashir, in power since he led an Islamist-backed military coup in 1989, advised opposition parties to prepare for the elections in 2020. His supporters arrived in packed buses for what was the largest rally in Khartoum in support of the president since the protests erupted.
On Tuesday during a visit to a military base in the railway city of Atbara, al-Bashir blamed violence during anti-government protests on conspirators, the state SUNA news agency quoted him as saying — conspirators who "planted traitors among us."
Atbara is a traditional bastion of dissent and one of several cities where anti-government demonstrations began Dec. 19, initially over rising prices and shortages but which quickly shifted to calls for al-Bashir to step down.
As al-Bashir spoke at the Khartoum rally, hundreds of protesters in Omdurman chanted, "revolution is the people's choice" and "freedom, dignity and justice."
Police used tear gas and fired in the air to disperse them before they could deliver their note to the legislature. There were no reports of casualties.
Sudan's Parliament is packed with al-Bashir's loyalists, who are campaigning to amend the constitution to allow the general-turned-president, already one of the longest serving leaders in the region, to run for a new term in 2020 elections.
SUNA reported in August that the country's ruling party has nominated al-Bashir for re-election in 2020.
Sudan's economy has stagnated for most of al-Bashir's rule. He has also failed to unite or keep the peace in the religiously and ethnically diverse nation, losing three quarters of Sudan's oil wealth when the mainly animist and Christian south seceded in 2011 following a referendum.
Critics say rampant corruption is eating up a significant part of government funds and engineering shortages of basic items to manipulate prices. The protesters have been chanting against the "government of thieves."
Authorities have said that 19 people died in the three weeks of protests, while Human Rights Watch said Tuesday at least 40 people have been killed since the protests erupted. Sudan's Interior Minister Ahmed Bilal Othman told lawmakers on Tuesday that police have arrested 816 people.