The most enduring obstacle to Mr. Trump is Mrs. Clinton’s formidable position in the Electoral College, where Democrats and a significant cohort of Republicans believe she is close to locking down the 270 electoral votes required for victory.

Early vote totals in several states, including North Carolina, are outpacing Mr. Obama’s performance at this time four years ago and have buoyed Democratic confidence even amid a burst of late adversity. And Republicans privy to private polling data said surveys they had seen since the news from the F.B.I. on Friday still showed Mrs. Clinton leading in North Carolina, a state Mr. Obama lost in 2012.

Facing a wall of opposition in Pennsylvania, Colorado, Virginia and New Hampshire, states Mr. Trump had hoped to dislodge from the Democrats, the Republican candidate has been forced to try to find other states he can win in the campaign’s final week. On Sunday, he was in New Mexico, claiming without evidence that the race is tied in the heavily Hispanic state, and on Monday, he appeared in Grand Rapids, Mich., vowing victory in a state no Republican has won since 1988.

“When you look around your state and you see the rusted-out factories, the empty buildings and the long unemployment lines, remember Hillary Clinton did much of this to you,” Mr. Trump said, invoking the names of the Big Three automakers and bringing Bob Knight, the storied former Indiana University basketball coach and Big 10 icon, onstage to introduce him.

Mr. Trump’s blue-state gambit was reminiscent of Mitt Romney’s late effort four years ago to make forays into a handful of Democratic-leaning battlegrounds, including Pennsylvania and Minnesota, and crack the political firewall assembled by Mr. Obama’s campaign. Mr. Romney did not succeed.

Russ Schriefer, an adviser to Mr. Romney’s campaign, said Mr. Trump was engaging in the “wishful thinking” of a candidate cornered on the electoral map.