OPEC+ oil producers may be forced to curb oil output even more due to rapidly filling storage capacity and plummeting demand due to the COVID-19 pandemic.

With crude consumption collapsing, the Organization of the Petroleum Exporting Countries, Russia and other producers, a group known as OPEC+, is due to implement a deal to cut supply by a record 9.7 million barrels per day (bpd) from May 1.

But that unprecedented deal to withdraw about 10 percent of global supply already may prove to be inadequate as demand has plunged by as much as 30 percent and the world's storage space is fast running out.

Vopak, the world's biggest independent storage company, said on Tuesday its tanks were almost full.

Tanks at Cushing, the delivery point for the U.S. crude futures contract, might not yet be full but any available space was already booked, analysts and traders said.

"We have to cut down, ... with or without OPEC output cut deal," Premium Times newspaper quotes Mele Kyari, the head of Nigeria's state-owned oil firm NNPC Group.

He said the West African state would have to cut production because it was hard to find any storage facilities for the oil.

Reuters also reports an OPEC source to say it was "logical" to expect the market to force more cuts on OPEC+ producers.

"We are in uncharted territory. Everything is possible, including the unbelievable," Reuters quotes an OPEC+ source.

In the meantime, OPEC is already considering further steps, barely two weeks since forging its last deal.

Saudi Arabia said on Tuesday it was ready to take extra measures along with OPEC+ allies and other oil producers, and Iraq echoed that position, although Russia was more cautious.