COLUMBUS, Ohio (AP) _ Democrat Howard Metzenbaum is counting on a third term as an outspoken consumer advocate in the Senate despite a well-financed and sometimes highly negative challenge from the mayor of his native Cleveland.

So far the polls are behind Metzenbaum, who has a reported $10 million bankroll to back up his re-election effort. His Republican rival, Cleveland Mayor George Voinovich, scoffs at the double-digit leads and says he'll benefit from a grassroots effort in all 88 Ohio counties in the final days and George Bush's presidential coattails.

Voinovich's campaign indicates he will have spent up to $8 million, but possibly as much as $10 million, by Nov. 8 on what has been largely a battle of television commercials, selected pot shots, and what some newspapers have called unprecedented sleaze on television.

The campaign is being fought over a variety of issues, and each has charged the other with changing past positions for political expediency.

Metzenbaum concedes that he has changed his position on capital punishment, from outright opposition to favoring the death sentence for drug pushers, who he said can afford competent lawyers. Metzenbaum voted for such penalties in the Senate drug bill.

Metzenbaum denied Voinovich's charge that he changed because polls show most Ohioans favor capital punishment.

Metzenbaum says Voinovich called for cuts in Social Security three separate times before announcing he was against any reductions.

Metzenbaum was angered by Voinovich television commercials that accused him of being soft on child pornography. On Oct. 10, he withdrew his summer-long promises for a fall debate, saying, ''The gutter is not a fitting place for a debate.''

Voinovich claimed in the ads that Metzenbaum used his position on the Senate Judiciary Committee to block a measure that would have dealt harshly with child pornography.

Metzenbaum, however, said he was concerned about a provision in the legislation that he claimed was unconstitutional and that the Justice Department concurred with his position. Metzenbaum later supported the bill after the provision was revised.

Voinovich said Metzenbaum is afraid to discuss his record. But he admitted that the commercials, which drew criticism from several major newspapers, produced an unexpected ''boomerang effect.''

He insisted his allegations were true but said his campaign should have set out to prove the charges in another way.

Voinovich, though mayor of Cleveland since he defeated Dennis Kucinich less than a year after the city went into default in 1978, also dealt with statewide issues as lieutenant governor to James A. Rhodes.

As mayor he presided over Cleveland's fiscal recovery, and as past president of the National League of Cities he often criticized the Reagan administration's urban policies.

Still, Voinovich is borrowing lines from Reagan and benefiting from the president's visits to Ohio. On the stump, Voinovich, 52, charges that the 71- year-old Metzenbaum is a liberal who is ''out of touch with Ohio's mainstream,'' having voted for higher taxes and cuts in defense spending while championing left-wing causes and opposing legislation to help industry create jobs.

''Two-thirds of his bankroll comes from out of state,'' Voinovich has charged.

Most of Metzenbaum's speeches and ads center on his record and claims of numerous accomplishments that include safer baby formulas, help for victims of rare diseases, and a law requiring 60 days' notice of plant closings.

On the attack, Metzenbaum contends Voinovich waffled over the plant closing law, once vetoed by Reagan, and that the mayor has given tacit approval to right-wing groups that have tried to paint the incumbent as a communist sympathizer.

Voinovich said he has asked the most visible of those groups, a Goodbye Metzenbaum Club that raises money and sends mailings from Cincinnati, to back off. But he said the club had persisted and that by competing with him for money, ''They have actually hurt me.''

Metzenbaum, responding to Voinovich charges that he is a pawn of labor, described himself as ''fiercely independent'' in recurring battles against corporate lobbyists and pressure groups that he said are pouring money into the Republican's campaign.

''There isn't a hit list anywhere between New York and California that doesn't have Howard Metzenbaum at the top, '' the senator said.

Metzenbaum's committee assignments include Energy and Natural Resources, Judiciary, Labor and Human Resources and the Select Committee on Intelligence.