A majority of District residents think Pepco did an unsatisfactory job restoring power after June’s violent storm, according to a Washington Post poll. But there is little consensus over a much-discussed solution to widespread power outages: burying power lines.

Asked how much extra they would be willing to pay for electricity in order to put more lines underground to protect them from severe weather, just over a third of those city residents who were surveyed said they would not be willing to pay any additional cost. Slightly fewer said they would be willing to pay as much as $10 a month, but less than one-fifth said they would be willing to pay more than that.

The unwillingness of rate-payers to substantially contribute complicates the debate over putting more lines underground, an idea that has gained favor in the weeks since the June 29 derecho storm but that remains costly.

Mayor Vincent C. Gray (D) on several occasions has called buried lines a “game-changer” that could reduce or eliminate the impact of severe weather on the power grid. The region saw prolonged outages not only after June’s storm but also after other severe weather events, including Hurricane Irene and a run of massive snowstorms.

A 2010 study done for the D.C. Public Service Commission estimated that burying all of the city’s aboveground power lines would cost nearly $6 billion. Spread out among all D.C. ratepayers over a 30-year period, Pepco has estimated that such an effort could add more than $100 a month to the average bill.

Less drastic measures, such as burying only primary lines known as “feeders,” would be significantly less costly, but poll results show that even a modest increase in power prices would be a tough sell.

Lenya Gregory-Perkins, 51, said she thinks power lines should be buried, but not at the expense of Pepco ratepayers.

“As a business owner, there’s something we have to invest in,” she said, referring to her hair salon. “We can’t expect the client to bear the brunt of the cost. . . . [Pepco] is able to enjoy their profit, and we don’t benefit from that.”

Among the 82 percent of respondents who registered an opinion, the average amount they would be willing to pay is $11 extra a month, with about half saying they’d pay more than $2, and half saying less than that. Among the 47 percent willing to pay something, the average rises to $20.

Nancy McNabb, 61, did not lose power at her Capitol Hill home in June, but she said she would still be willing to pay extra — as much as $10 a month — to bury lines elsewhere in the city.

“I think it’s an embarrassment that our nation’s capital doesn’t have better infrastructure,” she said. “The initial cost is nothing compared to all of these events that require a response with mutual aid from far away. That has to be very costly, and we know these events are becoming more frequent.”

Michael Horne, 43, who lost power for two days at his home in Southeast Washington’s Garfield Heights neighborhood, agreed that Pepco should rely less on out-of-town crews: “People need work. You have contractors right here.” But he was wary of higher power bills.

“Pepco’s already got a whole lot of money,” Horne said. “And a whole lot is falling on the taxpayers right now in this time of recession.”

Gray has moved to appoint a task force co-chaired by City Administrator Allen Y. Lew and Pepco chief executive Joseph Rigby to explore burying lines in detail. The group is expected to start meeting within weeks, said Pedro Ribeiro, a spokesman for Gray. One of the group’s objectives is to examine the cost and identify funding streams, such as government money or a dedicated surcharge on power bills.

A $2 addition to the typical monthly electric bill in D.C. could generate about $6 million yearly from the city’s 252,000 residential households; an $11 charge could raise about $33 million — over 30 years, that would approach the $1 billion-$1.5 billion estimates for modest citywide burial plans.

Wealthier residents, perhaps unsurprisingly, are more willing to pay for underground lines. Nearly half of those with incomes under $50,000 a year say they would not pay extra, while about a fourth of those making more than $100,000 yearly feel the same.

Those identifying themselves as politically liberal are almost twice as likely to be willing to pay extra than conservatives. Those under 50 are more likely to support an additional charge than those who are older.

More than 70 percent of the city’s power lines are already underground, according to Pepco statistics. Indeed, only a third of poll respondents say they lost power in the June storm. But many of those who kept their power are still critical of Pepco.

Those who live east of the Anacostia River were among the most likely to have lost power during the storm, with half of the residents there saying they lost electricity. But they are also among the most likely to approve of Pepco’s response — 44 percent rated it excellent or good.

About a third of those living west of Rock Creek Park said they lost power; in other parts of the city, about a quarter of residents did. Fewer than four in 10 residents in those areas rate Pepco positively.

African Americans, lower-income residents and those over 40 tend to give Pepco its best marks for power restoration. But at least as many in each group give Pepco negative marks.

Jon Cohen and Peyton M. Craighill contributed to this story.