President Joe Biden sought to quell fears of a wide-ranged banking collapse Monday in remarks given hours before U.S. markets opened and investors responded to the closure of Silicon Valley Bank and Signature Bank.

"Americans can rest assured that our banking system is safe, your deposits are safe," Biden said Monday morning, just 30 minutes before the New York stock market opened for the first time since federal regulators closed two significant banks.

After Silicon Valley Bank collapsed last week, New York-based Signature Bank was shut down by U.S. regulators on Sunday. Officials said that it was part of an action aimed at curbing the broader fallout from the failure of SVB.

Silicon Valley Bank now stands as the second-largest bank failure in U.S. history, and despite Biden stepping out to instill confidence, some experts continue to warn of possible "contagion."

"We want to make sure that the troubles that exist at one bank don't create contagion to others that are sound," U.S. Treasury Secretary Janet Yellen said during an interview with CBS Sunday.

Biden promised to do "whatever is needed" to prevent any additional bank runs, which is important because of the wide array of effects such events incur. Both economically and politically, the Biden administration must balance the line of ensuring that the shuddered Silicon Valley Bank does not have an outsized effect while not appearing to serve them a full government bail-out.

On the potential of a bailout, Biden made clear Monday that typical U.S. citizens would not be the losing party from last weekend's events. The money, instead, will come from the fees that banks pay to the Federal Deposit Insurance Corporation, Biden said.

"This is an important point: no losses will be borne by the taxpayers," Biden said on Monday morning. "Let me repeat that: No losses will be borne by the taxpayers."

Biden added that the bank's leaders will face repercussions, and promised investigations to hold those responsible for the collapse accountable.

"The management of these banks will be fired," he said. "If the bank is taken over by FDIC, the people running the bank should not work there anymore."

The Biden administration also announced plans to deter a situation like this from repeating, saying he would ask Congress and federal regulators to strengthen rules once weakened during the Trump presidency.

On Sunday, Biden directed Treasury Secretary Janet Yellen and National Economic Council Director Lael Brainard to coordinate regulators' responses to the banking crisis, saying the agreed-upon solution "protects workers, small businesses, taxpayers and our financial system."

U.S.