First Republic had been seen as one of the banks most at risk of failure, after a series of bank collapses raised fears of a crisis in the sector.
The Reuters report said that the government's involvement is helping bring more parties, including banks and private equity firms, to the negotiating table. But it is still unclear whether the US government is considering participating in a private-sector rescue of First Republic.
Wall Street banks have been trying to find a solution for First Republic since 11 of the biggest US lenders deposited $30 billion at the bank on March 16 to prevent a regional banking crisis that led to the failure of Silicon Valley Bank and Signature Bank.
Problems in the banking sector started to surface in the US last month when Silicon Valley Bank, which was the country's 16th largest lender, collapsed in the biggest failure of a US bank since 2008.
That was followed by the failure of New York's Signature Bank.
Authorities stepped in to guarantee deposits beyond typical limits in an effort to head off further runs on bank deposits.
In Europe, Swiss officials also brokered a rescue for troubled banking giant Credit Suisse, which saw 61.2bn Swiss francs ($69 billion) leave the bank in the first three months of the year.
The 167-year-old Credit Suisse Bank came to the brink of collapse before Zurich-based rival UBS rode to the rescue with a merger engineered and bankrolled by the Swiss authorities last month. UBS agreed to buy Credit Suisse for 3 billion Swiss francs ($3.3 billion), a fraction of its earlier market value.
Under the Credit Suisse UBS merger deal, holders of Credit Suisse AT1 bonds will get nothing, while shareholders, who usually rank below bondholders in compensation terms, will receive $3.23 billion.
(With inputs from Reuters)