ALSIP, Ill. — At least his home state is one of the places where Barack Obama is still welcome on the campaign trail.

Gov. Pat Quinn of Illinois has thrown open his arms to the president, who told a crowd of 6,000 recently, “I care about what happens here.”

Governor Quinn desperately needs black voters, who still adore President Obama, to turn out in force for him on Election Day. For his part, the president wants to avoid the embarrassment of losing deep-blue Illinois.

But despite Mr. Obama’s support and some positive economic news this year, Governor Quinn is running no better than neck-and-neck with a wealthy political novice, Bruce Rauner. Promising to use his business skills to tame state government, Mr. Rauner has poured $26 million of his fortune into tapping voter fatigue over high taxes and one-party rule in Springfield, the state capital.