President Obama is threatening Mitt Romney’s best route to victory in the electoral college, grabbing a significant lead over his Republican challenger in Ohio and a slender edge in Florida, according to two new polls by The Washington Post.

Democratic candidates have even larger leads in the Senate races in those two states, according to the polls. Together, the results suggest that with six weeks left until the election, Democrats hold significant advantages in some important battlegrounds.

In the presidential race, Obama is ahead of Romney in Ohio by 52 percent to 44 percent among likely voters. In Florida, the president leads 51 percent to 47 percent, a numerical edge but not a statistically significant one. Among all registered Florida voters, Obama is ahead by nine percentage points.

Romney began a two-day bus tour of Ohio on Tuesday afternoon in an effort to build enthusiasm for his candidacy and narrow the gap with Obama. The trip will take him from Dayton to Columbus to Toledo before he departs for Virginia, another key battleground where he is behind in public polls.

Rich Beeson, Romney’s political director, brushed aside the public polls of Ohio, telling reporters traveling with the candidate Tuesday that the campaign is making strategic decisions based on its internal surveys and research and remains confident about the outcome.

At his rally north of Dayton, Romney stressed that there are big philosophical differences between him and Obama, saying that the president’s approach is “foreign to anything this country has ever known.” Campaign advisers said they will seek to deliver a clear and consistent message outlining Romney’s policy priorities as a way to draw contrasts with the incumbent.

Romney has only a few days to drive that message before both campaigns are consumed by the presidential debates. The three presidential and one vice-presidential debate will offer Romney and his running mate, Rep. Paul Ryan (Wis.), their best chance to change the dynamics of the race and put Obama on the defensive. The first presidential debate will be held Oct. 3 in Denver.

With Romney lagging, Republicans face additional challenges down-ballot in the same battleground states. In the new Post surveys, Democratic Sen. Sherrod Brown of Ohio holds a substantial lead over Republican Josh Mandel, giving Democrats some breathing room in a race in which outside groups have put nearly $20 million toward defeating the incumbent. Brown leads Mandel 53 percent to 41 percent among likely voters.

In Florida’s Senate race, incumbent Bill Nelson (D) holds a 14-point advantage over Rep. Connie Mack (R), leading 54 percent to 40 percent among likely voters.

The new numbers come one week after a Post poll in Virginia showed Obama with a clear lead there. More than half of all the money spent in the campaign has focused on these three states, and many analysts say Romney has to win two of the three to capture the White House.

The past few weeks have been difficult for the Romney campaign, and the nominee’s advisers vowed to hit the reset button this week. But with the first debate scheduled for next week, Romney is under new pressure to refocus his campaign.

The new polls add to the evidence that Obama has benefited most from the two parties’ conventions, a series of sharp long-distance exchanges and a barrage of television ads. Nationally, polls continue to show a close race, but with newfound momentum for Obama in the battleground states that are likely to decide the election.

There would be few plausible ways for Romney to win the election if he lost Florida and Ohio, and even losing one of them would make his path to victory exceedingly narrow. No Republican has captured the White House without winning Ohio, and Florida, with its 29 delegates, may be even more vital to Romney’s hopes.

Both campaigns had thought of Florida as potentially more hospitable to Romney than to the president. But Obama’s competitive standing there — he is benefiting, as he did in the Virginia poll, from a huge lead among female voters — spotlights his challenger’s recent struggles.

For its part, Ohio has been the scene of hard-fought campaigns the past three elections and is widely considered a barometer of economic stress. Obama’s lead in the state is built in part on generally positive assessments of his job performance and on head-to-head comparisons with Romney on a series of issues. Slightly more than half of all Ohio voters, 53 percent, give Obama positive marks on his handling of the economy, with more, 56 percent, approving of his overall performance.

Thirty-six percent of all Ohio voters say they have been contacted by the Obama campaign; 29 percent say they have been contacted by the Romney side.

Fifty percent of all voters say they trust the president more to deal with the economy; 43 percent say so of his Republican challenger. By a much wider margin, 57 percent to 34 percent, registered voters in Ohio say Obama rather than Romney better understands the economic problems people are facing. The president also holds a big lead over his rival on who is trusted to advance the interests of the middle class.

There is far less difference, however, in the confidence voters express about whether the economy would improve more rapidly under a second Obama administration or a Romney White House.

The federal bailout of the automobile industry has been the focus of considerable debate between the candidates when they have touched down in Ohio. The poll shows that nearly two-thirds of Ohio voters say the loans that went to General Motors and Chrysler were mostly good for the state’s economy.

Still, most voters in Ohio say the economy is in bad shape. Yet even those people do not entirely blame Obama, with just under half of them saying the bad economy is his fault — about the same as the number who point the finger at the state’s Republican governor, John Kasich.

Just 38 percent of Ohio voters rate the state’s economy as “excellent” or “good.” Among those who see things positively, most, 68 percent, give Obama at least some credit for it. Nearly as many, 59 percent, credit Kasich.

In Ohio, Obama holds double-digit leads over Romney as the one earning more voter trust on five other issues — Medicare, Medicaid, taxes, social issues and international affairs — and he is numerically ahead on two others. Romney’s best issue is the federal budget deficit; on this, the two candidates run about evenly among all voters, and Romney has an apparent edge among those most likely to vote.

Obama benefits in both Florida and Ohio from double-digit advantages among female voters. In Ohio, male likely voters split about evenly between the president and his challenger. In Florida, 53 percent of men back Romney, and 45 percent support Obama.

In both states, Romney has the edge among white voters, while Obama wins 91 percent of nonwhite likely voters in Ohio and 74 percent in Florida.

Obama’s approval ratings in Florida, like those in Ohio, put him above the critical 50 percent threshold. Overall, 55 percent of Florida voters give him positive marks as president, and 52 percent say they approve of his handling of the economy.

One potential opportunity for Romney in the Sunshine State is that he runs about evenly with the president when it comes to who is trusted to handle the economy. But, by 60 percent to 35 percent, Florida voters say they trust Obama rather than Romney to advance the interests of the middle class. By 14 percentage points, they side with the president as the one with greater empathy toward people’s economic problems.

Medicare, the government’s health-care program for the elderly, was the focus of sharp debate in the weeks after Romney selected Ryan as his running mate. Today in Florida, the president runs 15 percentage points ahead of his challenger on whom voters trust more to determine the future of the program.

Peyton M. Craighill, Scott Clement, Aaron Blake and Sean Sullivan contributed to this report.