She said the chief election commissioner must think about it in wake of doubts being raised from several quarters.
"If BJP leaders consider themselves honest, why are they afraid of conducting future elections through ballot papers," she told a press conference on her 62nd birthday, alleging that the BJP came to power in Uttar Pradesh and Uttarakhand because of alleged faults in the functioning of the EVMs.
The Election Commission has rejected the charge.
Following the Gujarat Assembly polls in December, the EC did a random vote count on EVMs and paper trail slips in 182 polling stations in the state for greater transparency.
An official of the election machinery in the state had then said there was a "100 per cent match" in the vote count.
Mayawati, whose birthday is being celebrated as 'Jan Kalyankari Diwas' by her party, also expressed the possibility that the Lok Sabha polls, due in 2019, may be held this year along with various state Assembly elections.
She released the 13th edition of a book based on her life - A Travelogue of my Struggle-Ridden Life and BSP Movement-13.
"The BSP was the first party in the country to raise the issue of EVM anomalies soon after the results of the UP Assembly elections were out, and approached the Supreme Court. Later, whatever improvements were made in the EVMs, it was due to the efforts made by the BSP," she claimed.
The BJP won the Uttar Pradesh Assembly polls - held in February-March last year - after a gap of 15 years during which regional parties like the SP and the BSP had held sway.
Attacking Prime Minister Narendra Modi, Mayawati said: "'Har-Har Modi, Ghar-Ghar Modi' had a narrow escape of being thrown out from Gujarat this time. Had the voteshare of Dalits in Gujarat been 18 to 20 per cent, he would not had been able to escape."
The slogan she referred to was a campaign highlight of the BJP during the general elections in 2014.
Levelling allegation at the Centre, she said: "The BJP government wants to change the Constitution and laws. A minister in their government is talking about changing the Constitution, but no action has been taken against him."
The BSP chief alleged that the Congress and the BJP have damaged every section of the society after Independence.
"Today there is an atmosphere of communalism and casteism in every state. Capitalistic-minded political parties do not like our party. The BJP is trying to finish our party. Both, the Congress and the BJP are chor-chor mausere bhaai (hand in glove with each other). These parties do not want to see Ambedkar-inspired party (the BSP) to move ahead. First, it was the Congress and company, which tried to finish us, and now it is the BJP and company which is trying to do so," she said.
Mayawati said as she was not allowed to speak in the Rajya Sabha, she resigned.
"In the same manner, Babasaheb Bhimrao Ambedkar was also harassed, and subsequently he resigned from the post of the Union law minister," she said.
The BSP chief said: "I want to know from the Congress why Babasaheb was not honoured with Bharat Ratna. Why the recommendations of Mandal Commission were not implemented. The BJP government is virtually deactivating the reservation system, thereby rendering people jobless."
She said people who had to face trouble due to demonetisation would be given help by her party.