July 7 (Reuters) - Federal Reserve officials continued to discuss in June how they could potentially structure a permanent facility for supporting U.S. money markets, according to minutes from the central bank"s last policy meeting released on Wednesday. A "substantial majority" of Fed policymakers reiterated their support for such a program, which would allow eligible financial institutions to borrow cash on a short-term basis as needed, saying "the potential benefits of such a facility outweighed the potential costs," the minutes from the June 15-16 meeting stated. Several participants said the facility must be positioned as a "backstop" for markets, while some said it was important not to charge a rate so high that the program would be stigmatized. In one plan presented to Fed officials, the facility would charge a minimum of 0.25% to firms borrowing cash overnight, the top of the target range for the Fed"s overnight benchmark interest rate. Under that approach, the facility would be open to primary dealers and then expanded later to include banks that were interested. Several participants in the discussion said it could be appropriate to adjust the rate over time based on the economy or market conditions. The Fed began intervening in money markets in the fall of 2019 after reserves in the banking system fell too low, leading to a spike in short-term borrowing costs. Money markets were rocked again in March 2020 when the coronavirus pandemic led to a dash for cash, requiring the Fed to increase its repo offerings. In April, Fed officials discussed how having the support available through a permanent facility could allow the central bank to automatically respond to market pressures. Over the past several months, however, the Fed has been dealing with the opposite issue - an excess of cash in the banking system. read more Use of the Fed"s reverse repo facility, which gives firms a place to temporarily park cash with the central bank, reached a record of $992 billion on June 30.