Some member states believe the proposed €750bn ($857bn; £680bn) package is too large and should come as repayable loans, rather than grants. The talks are reported to have been testy, with tempers flaring at times. In the early hours of Monday morning, French President Emmanuel Macron reportedly banged his hand on the table and threatened to walk out of the discussions.
And Dutch Prime Minister Mark Rutte has admitted leaders were "close to failure" and talks could still "fall apart". Discussions are due to resume at 14:00 GMT on Monday for what is now the longest EU summit since Nice 2000 when talks lasted five days.
President of the European Council Charles Michel reminded the leaders that more than 600,000 people had died of the virus worldwide, and he hoped that the "headline tomorrow is that the EU has accomplished mission impossible".
Member states are split between those hit hardest by the outbreak, and those concerned about the costs of the recovery plan. Some northern nations like the Netherlands and Sweden have balked at the package, arguing it should take the forms of loans not grants.
But nations including Italy and Spain are desperate to revive their shattered economies, and have accused the EU of not doing enough to help countries hit by the pandemic.