WASHINGTON — Ending three straight days of solid opposition, Senate Republicans agreed Wednesday to start a full-scale debate on the most far-reaching proposals to toughen regulation of financial institutions since the Great Depression.

The Republicans’ decision cleared the way for a pitched battle over important details of the Democrats’ proposed overhaul. It also marked a milestone that substantially increases the odds that the sweeping legislation will be approved in the weeks ahead.

Tightening the reins on Wall Street and the financial industry is the top legislative priority for President Obama and congressional Democrats as they head toward November’s midterm elections. The House already has approved a version of the overhaul, and Obama has been campaigning vigorously for it.

Obama said Wednesday that he was “very pleased” the Senate legislation was moving forward.

“I want to work with anyone – Republican or Democrat – who wants to pursue these reforms in good faith,” he said during a visit to Quincy, Ill. “What I don’t want is a deal made that is written by the financial industry lobbyists. We’ve had enough of that.”

Two factors apparently led to breaking the legislative logjam: Republicans’ growing awareness of deep-seated public anger over Wall Street’s role in triggering the financial crisis, and a concession by key Democrats to change some controversial provisions.

Sensing that they were not likely to win a more comprehensive compromise, Republicans decided to stop negotiating and accept Democrats’ promises to address their most-touted criticism — that the bill does not do enough to prevent future taxpayer bailouts.

Differences over other aspects of the bill remained so wide that Republicans agreed to take those battles to the full Senate and suspend the delaying tactic that made some of their members increasingly uncomfortable.

“It is my hope that the majority’s avowed interest in improving this legislation on the Senate floor is genuine and the partisan gamesmanship is over,” said Senate Republican leader Mitch McConnell, R-Ky., who promised “many amendments.”

Democrats had accused Republicans of playing political games in blocking the start of debate on legislation crucial to preventing future financial crises. They cheered the decision to move forward. Senate Majority Leader Harry Reid, D-Nev., said he hoped it “foreshadows more cooperation to come.”

“I know Republicans have their own suggestions and amendments for improving this bill. So do Democrats,” Reid said. “Now that we’ll be able to begin that process, the American people will finally have the opportunity to watch and weigh those ideas.”

Republicans relented in the standoff amid new signs of financial turmoil, and reminders of old ones.

The financial markets have been reeling as the Greek government’s teetering finances threaten to destabilize the recovering world economy. And lawmakers and the public were riveted Tuesday by a hearing featuring executives of Goldman Sachs that highlighted some of the controversial practices on Wall Street that helped trigger the financial crisis.

“Look at this morning’s newspapers,” said Senate Banking Committee Chairman Christopher Dodd, D-Conn. “What more do you need to know about the importance of this issue and what’s going on in our nation?”

The legislation would create an agency to protect consumers in the financial marketplace; empanel a council of regulators to monitor the economy for signs of major risk; grant the government power to seize and dismantle teetering companies whose failure would pose a danger to the economy; and impose tough new regulations on complex financial derivatives.

Major financial companies and business groups oppose some key provisions, but there is broad support for much of the legislation.

“We support 80 percent of the bill and we encourage members to address remaining concerns around the 20 percent on the (Senate) floor,” said Scott Talbott, chief lobbyist for the Financial Services Roundtable, which represents large banks and other financial companies.

Earlier Wednesday, Senate Republicans remained united in voting for the third day in a row against a procedural motion to bring the bill to the full Senate. They said the move was not a stalling tactic, but designed to allow more time for bipartisan talks to produce a compromise.

But hours after that vote – which again led to charges by Democrats that Republicans were doing the bidding of Wall Street in blocking the legislation – the Republicans’ point person in the negotiations, Sen. Richard Shelby of Alabama, said the talks had reached an impasse. He had been negotiating for days with Dodd to reach an agreement.

“Chairman Dodd has assured me that he will address a number of concerns I have expressed with respect to ending bailouts,” Shelby said. “We have been unable, however, to make any meaningful progress on other important components of the legislation. It is now my belief that further negotiations will not produce additional results.”

Shelby and Dodd did not outline the specific concessions, but one is likely to involve revisions to a proposed $50 billion fund, paid in advance by financial institutions, to cover the cost should the government have to seize and dismantle a large financial company on the brink of bankruptcy.

Shelby said he still had problems with the proposed powers of one of the centerpieces of the bill, the new consumer financial protection agency, which he described as “a massive new bureaucracy” that would have “unchecked authority to regulate whatever it wants, whenever it wants, however it wants.”

Dodd said he would not agree to Shelby’s push to “weaken consumer protections” in the bill. But he said it was time for the full Senate to address those and other issues.