Both root for the Green Bay Packers. Both are charged with rallying white working-class voters. And with a debate a month away, Vice President Joseph R. Biden Jr. and Representative Paul D. Ryan are increasingly lobbing personal barbs while stalking each other across the same few Midwestern battleground states.


As Mr. Ryan buckled down on Sunday for his first full day of debate preparation, Mr. Biden campaigned over the weekend in Ohio, where he derisively labeled Mr. Ryan’s plan to overhaul Medicare “vouchercare.”


The vice president challenged the news media to “fact-check me.” The Romney-Ryan campaign responded with a direct round, singling out Mr. Biden. “Vice President Biden is once again advancing fabricated and disproven attacks,” said a spokeswoman, Amanda Henneberg.


On the campaign trail, the most thunderous applause for Mr. Ryan often comes when a local politician introducing him mentions the Oct. 11 debate with Mr. Biden. The Republican faithful are spoiling for the matchup, which they see as pitting Mr. Ryan, with his cold command of budget facts, against an outclassed Mr. Biden, whom they portray as gaffe-prone and too liberal for most voters.


Mr. Biden’s supporters say, in essence, bring it on. “I think the vice president can just tie him up in knots,” said former Gov. Ted Strickland of Ohio, who campaigned with Mr. Biden over the weekend. “I’m looking forward to the Ryan budget being discussed in the debates. I see Ryan as an empty suit.


“Far from being a deficit hawk, he’s gone along with nearly every budget-busting program the Republicans have supported, including massive tax cuts.”


Mr. Ryan and Mr. Biden, who are scheduled to be in Wisconsin on back-to-back days this week, emphasize their personal biographies to connect to blue-collar workers in Midwestern states hit hard by the financial crisis. Mr. Ryan’s hometown, Janesville, Wis., may now be best known for an auto plant whose closing he has pinned on the president.


Mr. Ryan speaks of “high school buddies” who lost jobs at the General Motors plant in Janesville, criticizing a promise in 2008 by President Obama, who was then a candidate, to help the plant retool and remain open.


Mr. Biden, speaking to General Motors workers last week in Lordstown, Ohio, said Mr. Ryan “wasn’t on the level.” He accused him of ignoring the fact that Mr. Obama’s auto industry bailout saved a million jobs, while Mitt Romney opposed it. “G.M. wouldn’t have been reorganized,” he said. “It would have been liquidated.”


Mr. Biden’s roots are in blue-collar Scranton, Pa., where, as he recalled last week at the Democratic National Convention in Charlotte, N.C., he lived with his grandparents when his father had to move away temporarily to find work.


“The financial crisis hit like a sledgehammer on all the people I grew up with,” he said in Charlotte.


Mr. Ryan’s selection by Mr. Romney put Wisconsin into play as a battleground state. At the same time, the congressman has reached out to Ohioans by suggesting that he is nearly a native son. He speaks of the “lucky buckeye” he carries, and he recently attended a football game between Ohio State and his alma matter, Miami University in Ohio.


Each candidate is his ticket’s bridge to the party base. Mr. Biden has spoken to labor groups as well as to Hispanics and African-Americans. For Mr. Ryan, his House budget proposal for $5.3 trillion in cuts over 10 years, a majority from programs for low-income Americans, has erased any ambivalence by the Tea Party wing of the Republican Party to embrace Mr. Romney.


The Ryan campaign is tamping down expectations by grass-roots supporters that the vice-presidential debate will be lopsided or easy. “Remember, the vice president has run for president twice,” a senior aide to Mr. Ryan said. “He’s probably one of the most experienced debaters in American politics right now.”


Vice-presidential debates often provide interesting theatrics, like Senator Lloyd Bentsen’s remarks to Senator Dan Quayle in 1988 that “You’re no Jack Kennedy,” but they rarely affect the outcome of the race.


Still, Mr. Ryan took time out from a fund-raising swing in Oregon on Sunday to meet with a small circle of advisers about the debate. One challenge for him will be to translate his sometimes overly technical policy explanations into ordinary language.


“I think it would be fair to say that after a number of years in Congress and putting together the budget, Chairman Ryan has an exceptional grasp of policy,” the aide said. “A lot of what we’re working on is the best way to debate, particularly given the vast experience the vice president brings to this process.”


Mr. Biden’s challenge in Ohio to the news media came after a recent assertion by a top Romney pollster that the campaign would not be “dictated by fact-checkers.”


“It’s amazing how they don’t like to be fact-checked, right?” Mr. Biden said in Zanesville, Ohio. “What they’re proposing will actually cause the Medicare trust fund that pays for the benefits when you go to the hospital, the doctor, to run out of money, a sufficient amount of money, by 2016.”


Mr. Biden issued the challenge: “I say to the press, ‘Fact-check me.’ ”


Although the vice president and Mr. Ryan are ideological opposites, they do have at least one thing in common.


Mr. Ryan took a break from debate preparation on Sunday to watch his favorite football team, the Packers, play its season opener. Mr. Biden revealed during a visit to Green Bay, Wis., last week that he, too, is a Packers fan. As a student in Catholic school, he said, the day began with a prayer invoking “the Father and the Son and the Holy Ghost and Vince Lombardi.”