• The move comes after a speech from Treasury Secretary Janet Yellen was released that said the government could backstop the deposits at more banks if there was risk of contagion.
• CNBC's David Faber reported Monday that JPMorgan Chase is giving advice on alternatives to the San Francisco bank that's drawn the attention of Wall Street for its large amount of uninsured deposits.

First Republic led a comeback rally in regional bank shares Tuesday, as investors hoped for some sort of strategic action by the troubled bank — or another big regulatory move — to stem the downward spiral in the sector.

The move comes after a speech from Treasury Secretary Janet Yellen was released that said the government could backstop the deposits at more banks if there was risk of contagion. Regional bank stocks have been under pressure since a large outflow of deposits led to the failure of Silicon Valley Bank and Signature Bank. Regulators guaranteed the deposits at those institutions after they were closed.

"The steps we took were not focused on aiding specific banks or classes of banks. Our intervention was necessary to protect the broader U.S. banking system," Yellen said Tuesday in remarks prepared for a speech to the American Bankers Association. "And similar actions could be warranted if smaller institutions suffer deposit runs that pose the risk of contagion."

First Republic shares soared 29.6% in Tuesday trading, following a 90% plunge so far in March and hitting a record low Monday. The SPDR S&P Regional Banking ETF gained 5.8%, following a 29% slide in March so far.

First Republic has been seen as one the remaining regional banks most at risk for the same fate as SVB, due to the large percentage of uninsured deposits it had as of the end of the fourth quarter. JPMorgan Chase led a group of 11 banks last week that deposited a combined $30 billion into First Republic, but its stock has continued to decline.

First Republic selected Lazard to help with a review of strategic options, The Wall Street Journal reported late Tuesday, citing people familiar.

"Following Thursday's uninsured deposit of $30 billion by the 11 largest banks in the country, together with cash on hand, First Republic Bank is well positioned to manage short-term deposit activity," First Republic said in a recent statement.

CNBC's David Faber reported Monday that JPMorgan is giving advice on alternatives to the San Francisco bank. Those alternatives include a capital raise or possibly even a sale, sources told Faber. CNBC's Kayla Tausche reported Tuesday that the capital infusion would come only if a sale fails to materialize.

Reuters reported on Tuesday that major bank leaders were having a pre-scheduled meeting in Washington, with First Republic as a topic of discussion, and that the regional bank was considering downsizing as a way to raise cash.

Also helping sentiment was a report by Bloomberg News that the Treasury Department is studying whether regulators have the authority to temporarily insure deposits above the current Federal Deposit Insurance Corp. cap without the approval of Congress, citing people with knowledge of the talks. Though, the report said these government officials don't believe such drastic action is necessary yet.

"There has been speculation that the limit could be doubled, and further speculation that the FDIC could decide to insure all deposits," wrote Alexander Twerdahl, a Piper Sandler analyst, in a recent note. "In actuality, it would take an act of Congress to change the FDIC's insurance limit and our understanding is that it isn't an issue that is likely to be taken up any time soon."

KeyCorp and U.S. Bancorp each jumped more than 8%, while Western Alliance and PacWest Bancorp. gained nearly 15% and more than 18%, respectively.

— CNBC's Michael Bloom contributed reporting.