Many eyes in his home state and Washington are on the Massachusetts Republican, who is in a political bind even though there is only a remote chance that his vote will prove to be pivotal, as it was in the 2010 Dodd-Frank Wall Street regulatory overhaul.

The bind is this: If Brown continues to oppose higher taxes on the rich, he risks undercutting the reputation he has tried to build as a Massachusetts moderate and a man of the people.

But if he approves the president’s plan to raise the tax rates on upper-income families, he could find himself accused of flip-flopping and out of favor with his Republican base.

The drama over whether Congress will strike a deal to avoid automatic tax increases and spending cuts before Jan. 1 is mostly playing out in the Republican-controlled House.

If Speaker John Boehner and the House GOP can deliver on a deal, it is likely the Senate will go along as well.

But Democrats are hunting high and low for Republicans like Brown in either chamber of Congress who may be willing to loosen their opposition to tax hikes for the wealthy and send a signal to their fellow GOP members that compromises the correct course.

Brown, who boasted of his willingness to work with Democrats during his failed re-election bid, is seen as holding the potential to cross party lines.

Massachusetts Democratic Party chairman John Walsh Wednesday said Brown should now get in line with the message of the state’s voters, who chose two candidates, President Obama and Senator-elect Elizabeth Warren, who favored higher taxes on the rich.

“I hope the senator heard it. It couldn’t be clearer. He is there to represent us,” Walsh said.

How Brown handles the issue also carries great significance for him, personally, and for the reputation he has sought to build as a New England moderate. His ultimate position is likely to be the last major act of his Senate term and could define how he is perceived in liberal-to-moderate Massachusetts as he ponders his political future.

If he continues to stand against raising taxes on the wealthy, he risks bearing the brunt of blame with other Republicans if the fiscal cliff is breached.

A Washington Post-Pew Research poll published this week showed that a majority of Americans, by 53 percent to 27 percent, will point the finger at Republicans in Congress instead of Obama if a deal is not cut.

But reversing course also carries risks for Brown, because he signed a no-tax-increase pledge for antitax activist Grover Norquist’s organization, Americans for Tax Reform, and he campaigned as a defender of tax cuts for all during his campaign against Warren in November.

Switching position would basically be the same as embracing Warren’s stance, which he fought against vigorously during the campaign.

Thus far, Brown is refusing to say if he is willing to reconsider his opposition to raising tax rates for the wealthy.

He also has not commented publicly on the House Republican proposal to limit itemized deductions, although he has supported that approach in the past.

Brown did not respond to requests for an interview on his views Wednesday and his office did not respond to e-mailed questions about his positions. He told a Bloomberg reporter in a Senate hallway this week that he has not taken any position on possible fiscal cliff compromises, because there are no proposals yet before the Senate for him to consider.

“I’m going to vote the same way I’ve always voted,” Brown said. “I’m going to finish like I started, which is I’m going to read the bills and see how they affect our state, our country, our debt, and our deficit, and vote.”

That is a typical response for Brown, who has established a track record of waiting until the last minute in most Capitol Hill debates before taking a stand.

In this case, however, even not taking a stand could be viewed as a bit of softening in his opposition to increased taxes for the rich. Brown said in September during the heat of the Senate race that his opposition to such a move was “crystal clear.’’

Now Brown is back in the Senate as a lame duck, and pondering a different sort of political future.

“He may be a lame duck, but he can still quack,’’ said Massachusetts Republican strategist Todd Domke. “If he speaks out now, he could have an impact. He could be a catalyst for compromise.