White House predictions about how the US economy might rebound from the coronavirus crisis and how quickly a vaccine might be rolled out came under question on Sunday. The US will need more tests before schools can reopen later in the year, said Lamar Alexander, Republican chairman of the Senate health committee. Appearing on NBC’s Meet the Press, the Tennessee senator appeared to question the White House’s ability to meet a target of having 100 million vaccine doses by autumn and 300 million by the end of 2020. Alexander called it “an amazingly ambitious goal” and added: “I have no idea if we can reach that.” No vaccine has been approved though a number are under development. Neal Kashkari, president and chief executive of the Federal Reserve Bank of Minneapolis, told ABC’s This Week he would welcome a robust economic recovery. “But that would require a breakthrough in vaccines,” he said, “a breakthrough in widespread testing, a breakthrough in therapies, to give all of us confidence that it’s safe to go back. I don’t know when we’re going to have that confidence.” White House advisers said they had begun informal talks with Congress about what to include in another round of coronavirus relief legislation. But they also predicted further jobs losses. Treasury secretary Steven Mnuchin and economic adviser Larry Kudlow said they were holding discussions with lawmakers on issues including aid to states whose finances have been devastated by the pandemic. Another economic adviser, Kevin Hassett, said legislation could include food aid and broadband access. Since March, Congress has passed bills allocating $3tn to combat the pandemic, including taxpayer money for individuals and companies to blunt an economic impact that includes an unemployment rate of 14.7% in April, after job losses unseen since the Great Depression of the 1930s. Democrats who control the House of Representatives are moving to unveil new relief legislation as early as this week. But the White House is in no hurry. “Let’s take the next few weeks,” Mnuchin told Fox News Sunday. “We just want to make sure that before we jump back in and spend another few trillion of taxpayers’ money that we do it carefully,” he said. “We’ve been very clear that we’re not going to do things just to bail out states that were poorly managed.” Pressure may mount as the economic picture worsens. On CBS’s Face the Nation, Hassett said unemployment could rise “north of 20%” in May or June before what administration officials insist will be a robust recovery. The April unemployment rate undercounts some out-of-work Americans, economists say. Asked if the country could be facing a “real” rate of close to 25%, Mnuchin said: “We could be.” Such a rate also includes people who have lost jobs and are not actively seeking employment and people considered underemployed. Democrats are pushing for another massive bill that would include more money for state and local governments, coronavirus testing and the US Postal Service. Advisers say the White House will not consider new stimulus legislation in May. “It’s not that we’re not talking. We are. It’s just informal at this stage,” Kudlow told ABC. “We’re collecting ideas for next steps, which will undoubtedly be data-driven.” Kudlow said he took part in a Friday call with House lawmakers from both parties, and plans to do the same on Monday with members of the Senate, which is held by the Republicans. “If we go to a phase-four deal, I think that President Trump has signaled that, while he doesn’t want to bail out the states, he’s willing to help cover some of the unexpected Covid expenses that might have come their way,” Hassett told CNN’s State of the Union. The White House is “absolutely” pushing for a payroll tax cut, Mnuchin said. Trump has called for a cut to the tax, which is paid by employers and workers and funds Social Security and Medicare. The proposal has little congressional support. Trump has also threatened to withhold funds from states that limit cooperation with federal immigration enforcement, a stance critics say would exploit a public health crisis to advance political goals.