The debt-laden country is struggling to pay for imports of fuel and other goods due to a foreign exchange crisis, leading to hours-long power cuts and a shortage of essentials, Reuters reported.
Widespread street protests have continued despite a weekend curfew.
"Considering this a national need, the time has come to work together for the sake of all the citizens and future generations," Rajapaksa's media office said in a statement.
"The president invites all political parties representing in the parliament to come together to accept ministerial portfolios in order to find solutions to this national crisis."
Central Bank of Sri Lanka Governor Ajith Nivard Cabraal said he had also offered to quit.
"In the context of all cabinet ministers resigning, I have today submitted my resignation as Governor," he said on Twitter.
The developments come after Rajapaksa declared a state of emergency on Friday, following protests over the economic hardship faced by the people of the country.
The Colombo Stock Exchange suspended trading twice on Monday morning due to a sharp fall in the benchmark share price index.
Traffic was back on the streets of the country's main city Colombo on Monday, but there were reports of sporadic and peaceful protests from across the country.
The island nation of 22 million, off India's southern tip, is also grappling with soaring inflation after the government steeply devalued its currency last month ahead of talks with the International Monetary Fund for a loan programme.
The country's expenditure has exceeded its income under successive governments while its production of tradable goods and services has been inadequate. The twin deficits were badly exposed by the COVID-19 pandemic that crippled its economic mainstay, the tourism industry.