Sen. Rand Paul on Wednesday reintroduced legislation to force a government watchdog audit of the Federal Reserve, and the bill has its best chance yet of finally clearing Congress.

Mr. Paul’s father, former Rep. Ron Paul, pioneered the push for the legislation during his time in the House, and turned it into a potent political weapon during his presidential campaigns in 2008 and 2012. The senator, who is eyeing a 2016 presidential bid, has picked up the banner.

“A complete and thorough audit of the Fed will finally allow the American people to know exactly how their money is being spent by Washington,” the younger Mr. Paul said in a statement. “The Fed’s currently operates under a cloak of secrecy and it has gone on for too long.”

His legislation has 30 sponsors, including Sen. Mazie Hirono of Hawaii, the sole Senate Democrat who has signed onto the bill as of now.

In a statement, Ms. Hirono praised the Fed for helping the economy recover from the 2008 slump.

“That said, we’re now far enough away from the financial crisis for a full and transparent review of the actions taken on behalf of American taxpayers,” she said. “Greater transparency will help us prevent another Great Recession, and I believe will increase the American public’s confidence in the Fed rather than diminishing it.”

A similar bill has cleared the House twice in recent years with overwhelming support, including about half of House Democrats. But Sen. Harry Reid of Nevada, who was majority leader of the Senate until his party’s catastrophic election losses last year, blocked Senate action.

New Majority Leader Mitch McConnell, a Kentucky Republican along with Mr. Paul, is a co-sponsor of the audit bill, and it is likely to receive a vote at some point.

Both Sens. Marco Rubio of Florida and Ted Cruz of Texas, Republicans also pondering White House runs, have joined as co-sponsors.

“The Fed has expanded its balance sheet fivefold, yet economic growth is still tepid, businesses are sitting on cash, and median income and household wealth are depressed,” Mr. Cruz said in a statement.

Federal Reserve Chair Janet Yellen has urged Congress not to approve an audit, arguing it could intrude on the board’s independence.

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