Chanting "One Army, One People," the protesters rallied in the capital's streets, following a call by organisers to march on the army headquarters, located near the residence of Bashir.
In a separate demonstration, protesters also reached the army building of the central town of Madani, witnesses told AFP by telephone.
The crowds chanted the catchcry of the movement "peace, justice, freedom," as they marched through the streets of Khartoum and reached the army building, which is also the defence ministry, according to onlookers.
Protests have rocked the east African country since December, with angry crowds accusing Bashir's government of mismanaging the economy that has led to soaring food prices and regular shortages of fuel and foreign currency.
On February 22, the veteran leader imposed a nationwide state of emergency to quell the protests after an initial crackdown failed to rein in protesters.
Since emergency rule came into effect, the demonstrations have been largely confined to the capital and its twin city of Omdurman, but organisers had called for widespread rallies and a march on the army headquarters on Saturday.
April 6 was chosen for the nationwide rallies as it was the day of a 1985 uprising that toppled the then regime of president Jaafar Nimeiri.
Before the protests began, security forces deployed in large numbers in key Khartoum squares and in Omdurman, across the Nile.
"There's a heavy security deployment where the protesters were to gather for the march, but they still came out and are chanting anti-government slogans," a witness told AFP without revealing his name for security reasons.
Witnesses said plainclothes security agents were preventing even passers-by from reaching downtown areas.
Shops and markets in downtown Khartoum were ordered shut before the march by security agents, according to onlookers.
"Those walking in groups were immediately detained or asked to return to their homes by security forces," another onlooker said.
In recent days, activists have been circulating leaflets urging residents to participate in Saturday's march, several residents said.
The protest movement was initially led by the Sudanese Professionals Association, but later several political parties including the main opposition National Umma Party threw their support behind it.
Analysts say the movement has emerged as the biggest challenge yet to Bashir's three-decade rule.
The veteran leader has remained defiant however, and has introduced tough measures that have seen protesters, opposition leaders, activists and journalists arrested.
Officials say 31 people have died in protest related violence so far, but Human Rights Watch has put the death toll at 51 including children and medics.
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