As many governors across the US say coronavirus testing is far from full capacity, a few states may lift restrictions this week.

Announcements of lifting restrictions come as the United States' coronavirus death toll topped 40,000 on Sunday, according to data from Johns Hopkins University. There were at least 40,680 deaths among more than 759,780 coronavirus cases, the university's Covid-19 tracker shows.

Several regional groups of states have discussed strategies to reopen with each other.

Other states are working on plans to reopen following the release of guidance from the Centers for Disease Control and Prevention last week.

The guidance suggests that states expand their testing capacity but some say they need more supplies and staff to do so.

States want testing capacities expanded

Over the weekend President Trump said governors across the country had the capacity to complete more testing but weren't utilizing their resources properly.

"They don't want to use all of the capacity that we've created. We have tremendous capacity," Trump said during a White House briefing. "They know that. The governors know that. The Democrat governors know that. They're the ones that are complaining."

Virginia Gov. Ralph Northam called the President's comments "delusional."

Northam said he wants to see at least two weeks of trending numbers of coronavirus cases decreasing before businesses in his state reopen.

"We want to make sure we're doing it responsibly and we're doing it safely," the Democratic governor told CNN's Jake Tapper on The State of the Union.

"We've been fighting for testing, it's not a straightforward test. We don't even have enough swabs believe it or not," Northam said.

Maryland Gov. Larry Hogan also told CNN's Jake Tapper that Trump's claim that governors have plenty of tests and simply aren't doing their jobs is "absolutely false."

Illinois Gov. J.B. Pritzker gave similar remarks Sunday. He said he wants things to be reopen so residents can enjoy recreation this summer but said they need to listen to scientists and doctors investigating the virus before they can comfortably and safely reopen businesses to the public.

In order to safely do so, the state needs more lab technicians, more reagent and more RNA extractors to analyze specimens taken, Pritzker said.

Harvard researchers warned that if the country wants the economy to open back up -- and stay that way -- testing must go up to at least 500,000 people per day.

Testing nationwide is currently at 150,000 per day, they said, adding that "If we can't be doing at least 500,000 tests a day by May 1, it is hard to see any way we can remain open."

Gov. Cuomo: 'We controlled the beast ... we did not kill the beast'

Aside from testing for the virus itself, antibody tests are being considered as a main component in determining reopening, Dr. Stephen Hahn, the FDA Commissioner, said in a statement Saturday.

"Results from these tests can help identify who has been infected and developed antibodies that may protect from future infection as well as identify those still at risk," the statement said.

As promising as these tests could be, the FDA statement said that it is still unknown if people with antibodies are fully protected from reinfection or how long their immunity might last.

Despite the lack of information on how effective the current antibody testing will be, New York Gov. Andrew Cuomo said Sunday that the state will undertake the "most aggressive" statewide antibody testing survey in the country.

"We are going to start antibody testing across the state (Monday)," Cuomo said. "We are going to do that in the most aggressive way in the nation. This will be the first true snapshot of exactly how many people were infected by COVID-19 and where we are as a population and will help us to reopen and rebuild without jeopardizing what we've already accomplished."

"We controlled the beast," Cuomo said. "We did not kill the beast. And the beast can rise up again."

If and when restrictions are lifted they will need to monitor infection and hospitalization rates and adjust their plan accordingly.

"And if we went through all of this and lost all of these people and forced essential workers and hospital workers to do unbelievable tasks to get us through this crisis and we recreate the crisis, then shame on us," Cuomo said.

Louisiana, which saw the fastest per capita growth of coronavirus than anywhere in the world a few weeks ago, seems to be "trending in a good direction," Gov. Bel Edwards said during a Sunday press conference.

While case counts continue to rise in the state, the rate of hospitalizations and patients on ventilators has decreased, according to Edwards.

"We're in much, much better place today than we thought we were going to be," Edwards said, saying Louisiana citizens are taking the stay at home order seriously and sticking to social distancing guidelines.

Illinois Public Health Director Ngozi Ezike shared a similar sentiment that despite more than 30,000 positives statewide, she was "cautiously optimistic" that the number of new cases would continue to decrease.

Downward trends in case counts are part of the CDC's guidelines to "get and keep America open" issued on Friday. The guidelines also recommend that rapid testing and increased hospital capacity be considered prior to states reopening.

Texas Gov. Greg Abbott announced three orders and the establishment of a Strike Force to Open Texas. The team will work to develop strategies, standards and time frames for reopening businesses. Abbott plans to announce a plan to reopen on April 27, according to the order.

Despite several state leaders announcing their intention to reopen, over the weekend many civilians held protests across the country demanding to be let back to work to try and resuscitate the economy.

More protests are planned for Monday in upstate New York, Pennsylvania, New Mexico and Missouri.

Over the weekend, protesters flooded Austin calling for the reopening of businesses. Similar protests were held in Michigan, Ohio, Kentucky, Minnesota, North Carolina and Utah last week.