WASHINGTON — After a month in which some prominent Democrats openly questioned President Barack Obama’s campaign strategy, the mood at the White House has risen, with strategists believing their efforts to define Mitt Romney as a corporate outsourcing specialist are proving a success with swing voters.

The shift can be seen in several recent public polls that have shown Obama ahead in key states and moving upward nationally.

In Gallup’s daily tracking poll, for example, Obama has taken a 48 percent to 44 percent lead over Romney, the first significant lead that either candidate has held since late April in Gallup’s survey.

The upward movement for the president, which has been sustained for six days, began before the Supreme Court’s decision upholding most of his signature health care law, so it does not reflect a reaction to that ruling.

Democratic strategists credit their attacks on Romney’s record at Bain Capital, the same record Romney touts as proof of his ability to fix what’s wrong with the U.S. economy.

The shift comes as Obama prepares to set off on his first multiday campaign swing this year, a two-day bus tour of Pennsylvania and Ohio beginning Thursday, complete with ice cream socials and events that are being billed as a “Celebration of the American Worker.”

Tougher times may be around the corner. Friday will bring the monthly jobs report, which could reflect continued weakness in the U.S. economy. The downturns in Europe and China have begun to affect American manufacturers, according to economic indicators released Monday. In addition, Romney and his allies continue to outpace Obama and the Democrats on fundraising.

Obama’s team believes his message about Romney is getting through to voters, and they plan to step up the intensity. While Obama offers a genial tour, the television spots and surrogates in battleground states will be pushing the message that Romney was an outsourcing “pioneer.”