Even as lawmakers in both chambers are racing this week to enact a massive coronavirus relief package — the third in as many weeks — House Democratic leaders are telling members to expect at least two more stimulus measures in the weeks and months ahead.

On a marathon conference call with the House Democratic Caucus Tuesday afternoon, House Majority Leader Steny Hoyer Steny Hamilton HoyerHouse Democrats postpone vote on marijuana decriminalization bill Democrats scramble on COVID-19 relief amid division, Trump surprise The Hill's Morning Report - Sponsored by The Air Line Pilots Association - Pence lauds Harris as 'experienced debater'; Trump, Biden diverge over debate prep MORE (D-Md.) told lawmakers there will likely be a fourth and fifth phase of pandemic relief, according to a source on the call.

Those would follow on the heels of an initial $8.3 billion package, which focused on the immediate health concerns surrounding the coronavirus, and a second $104 billion proposal, designed to ease the financial stress on those most directly affected by the fast-moving pandemic.

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The third phase remains in flux, as Senate leaders and White House officials are scrambling to secure a deal on a massive economic relief package that could exceed $2 trillion. The top negotiators said Tuesday afternoon that they're close to a deal, but partisan disagreements over several provisions — including funding for food stamps and the airline industry — still need ironing out.

It remains unclear what the next rounds of relief might entail. But House lawmakers are not alone in predicting that the economic fallout from the coronavirus will require Congress to take additional action.

Senate appropriators are already in the process of fielding recommendations for Phase IV, according to sources familiar with the talks.

It's also unclear when additional rounds of relief might arrive. The House is currently on recess, and if Speaker Nancy Pelosi Nancy PelosiAs families deal with coronavirus, new federal dollars should follow the student Sunday shows - Ruth Bader Ginsburg's death dominates Hypocrisy rules on both sides over replacing Justice Ginsburg MORE (D-Calif.) and the members of her caucus get behind the Senate's Phase III package, there's an expectation that it could move through the lower chamber by unanimous consent, precluding the need for lawmakers to return to Washington amid growing fears of travel and crowds.

And the Senate, after passing the mammoth stimulus, is expected to leave town for an extended recess. Some aides and lawmakers are predicting both chambers might not be back to the Capitol before the week of April 13 — or later, depending on the evolution of the virus.

As Phase III moves closer to the House, Pelosi said she's hoping it wins enough support from her caucus to move it through the lower chamber by unanimous consent. On Tuesday's conference call she outlined a number of Democratic victories in the ongoing Senate negotiations, including $150 billion for states and local governments, $130 billion for hospitals and the creation of an independent inspector general to oversee the distribution of $500 billion in loans to hard-hit industries.

But sticking points remain, not least a disagreement over increased funding for food stamps, which many liberal Democrats are insisting upon.