Warren, a Democrat, leads Brown, a Republican, 43 percent to 38 percent, a shift from the Globe’s last poll in May, when Brown held a 2-point lead. But the race remains within either candidate’s grasp, with 18 percent of voters still undecided, said Andrew E. Smith, the Globe’s pollster and the director of the University of New Hampshire ­Survey Center.

This survey is the sixth of eight public polls taken this month that show Warren ahead.

Warren’s lead is within the poll’s margin of error of plus or minus 4.4 percent, meaning a spread of as much as 8 percentage points between the candidates would still statistically count as a dead heat. Still, the survey is sobering for Brown six weeks before the election.

“It’s trending away from Brown,’’ said Smith. “Brown right now is not doing well enough among Democrats to offset the advantage that ­Warren has,” said Smith. “That’s just such a big obstacle to overcome for any Republican candidate” in Massachusetts.

Brown’s hurdle is that even as voters from both parties say they like him personally, regard him as bipartisan, and give him high marks on his job performance, many also say they plan to stick with their political party. And in a presidential year, with high turnout expected, that bodes especially well for Democrats on the ballot in Massa­chusetts. The party holds a 3-to-1 registration advantage over Republicans in the state.

In the presidential contest, the poll shows President Obama leading former governor Mitt Romney by 27 percentage points in Massachusetts, a 57 percent to 30 percent landslide. If that holds up, Brown will face a strong Democratic headwind that could overwhelm his individual ­efforts as a candidate in the Nov. 6 election.

Of those who said they will vote for Obama, just 9 percent said they also plan to vote for Brown, while 20 percent of Obama voters say they are still undecided in the Senate race, according to the poll.

Romney’s popularity in Massachusetts has plummeted since he won the 2002 governor’s race as a moderate Republican who favored legalized abortion and held other socially liberal stances.

The poll shows 33 percent of respondents view the GOP presidential nominee favorably, compared with 60 percent who viewed him unfavorably, a significant drop since a May poll showed a split among Massachusetts voters on that question. His poor showing in ­national polls could dampen Republican participation in Massachusetts.

Sampling for the poll of 502 likely voters began the day after the candidates’ first debate, running from Sept. 21 through Sept. 27. Throughout that period, both candidates advertised heavily. They also sparred over Warren’s controversial legal work on behalf of a coal mining company and over a video capturing at least one Brown staff member, along with several from the state Republican Party, engaged in a mock Indian war chant that was criticized as offensive.

After a year of seesawing polls that seemed to show an edge for Brown, a string of polls taken this month have suggested an advantage for Warren.

Charlie Brewer, a 53-year-old self-described independent voter from Franklin, responded to the Globe’s poll saying he was still undecided in the Senate race, though he said he was certain to vote for Obama.

An admirer of what he sees as the senator’s bipartisan spirit, Brewer is the type of voter Brown hopes to win over. But over the last week, since the poll was taken, Brewer said he has made up his mind, in large part because he was turned off by the incident with the Native American chanting. “If it were today, I think the image of those guys whooping with the tomahawk chop, I’d vote for Ms. Warren,” Brewer said.