BOSTON—As polls across Massachusetts closed at 8 p.m., both campaigns for U.S. Senate said they were cautiously optimistic heading into what could be a long night of ballot counting.

At the Fairmont Copley Plaza, upbeat supporters of Harvard Law School professor Elizabeth Warren began gathering in the lobby and in a red, white, and especially blue-themed ballroom for what they hope will be a celebratory night.

Early estimates indicate that voter turnout across the state is approaching a record high, which should help Democrat Warren, who has had a slight edge over Brown in recent polls. Warren’s campaign manager Doug Rubin said the first-time candidate had stopped at eight or nine polling stations throughout the day in a last-minute push to win over undecided voters.

“I feel like we’ve done everything we wanted to do from a campaign perspective. We’ve gotten a lot of people out and excited about Elizabeth,” Rubin said. “Now we’ve just got to wait and see.”

Three blocks away at the Boston Park Plaza Hotel, Brown supporters gathered around televisions watching early returns roll in, anticipating an evening that they hope will include a victory for Brown.

Though most attendees at the Brown party agreed that a high turnout would be favorable for Warren, Republican officials said that early reports were positive for Brown.

“By the early counts that we’ve seen, turnout has been extremely high just about everywhere. It’s extremely high in some of the suburbs,” said Tim Buckley, director of communications for the Massachusetts GOP. “The Route 128 belt, the South Shore, and up in the North Shore...those areas are extremely good for Senator Brown. In those areas he carried 65-70 percent of the vote in the special election.”

Buckley said that some reports suggest that turnout in major cities, where Warren is heavily favored, was not prohibitively high.

Even as they expect him to win, many of Brown’s supporters said that he is the underdog, outmatched by Warren’s significant ground operation and political advantage in the left-leaning Massachusetts.

“With the entire Democratic apparatus in Massachusetts...lined up against him, if [he] comes through and delivers on Election Day, I’d call it a bit of an upset,” said former Republican gubernatorial candidate Charles D. Baker ’79.

“But just because it’s an upset doesn’t mean it’s not what I expected to happen,” Baker added. “I always thought that Scott would get re-elected, because he deserves to be re-elected.”

Supporters in both ballrooms and at the Hynes Convention Center across town, where supporters of Republican presidential hopeful Mitt Romney are gathered, eagerly await the results from across the state and around the country.

—Staff writer Nicholas P. Fandos can be reached at nicholasfandos@college.harvard.edu