Donald Trump has done exactly what he promised: He has scrambled the election map, putting into play states that have been out of the Republican Party’s reach for a generation, but he also risks losing states that have long been GOP strongholds.

Since Mr. Trump chased the last of his rivals from the Republican presidential primary contests in early May, polling has shown him within striking distance of likely Democratic nominee Hillary Clinton in Michigan and New Jersey, and statistically tied with her in Pennsylvania. None of those states has voted for the Republican candidate in a presidential election since 1988.

Mr. Trump said the polls bear out his claim that he could bring a populist Republican message to voters who traditionally have tuned out the party. He has hit a sweet spot among Americans who are angry at Washington and who fear rising immigration rates and bad trade deals will leave them out of the new economy.

The billionaire businessman’s brash style plays well with some disaffected voters but could be harming the party with others. Some polling shows Mr. Trump with only slim leads in Georgia and Arizona, which have been reliably Republican.

States that have been battlegrounds — Virginia, Florida, North Carolina and Ohio — remain razor-close, according to polls conducted in the past month.

Patrick Murray, director of the Monmouth University Polling Institute, said the map shows many voters aren’t particularly happy with either choice, leaving the election outcome uncertain.

“We are going to see Democratic states with not a big Democratic advantage and Republican states with not a big Republican advantage, and purple states will stay purple,” he said. “That is because the dominant party in each state includes a significant number of voters that are not happy with the choice presented to them by their party’s nominee.”

Mr. Trump will need to find more fertile ground if the Republican Party is to recapture the White House.

In the early 2000s, the electoral map was solidly blue along the West Coast, the Northeast and the Upper Midwest, while the South and mountain regions were red.

Barack Obama captured new territory for Democrats in 2008 by using strong minority turnout and dissatisfaction with President George W. Bush to expand into the mid-Atlantic, push deeper into the Midwest and make gains in the mountain regions.

Mr. Trump has said he will reverse those gains by focusing on about 15 swing states and making efforts to win Democratic strongholds New York and California.

“We are going to play heavy, as an example, in California,” he said. “No other Republican, they wouldn’t even go to dinner in California. They wouldn’t do it.”

Polling suggests that winning California or New York is still a stretch. The Golden State Poll this week gave Mrs. Clinton a 12-percentage-point lead over Mr. Trump in California, and a Sienna poll put her up 21 points in New York.

Mr. Trump is likely to have better luck in the industrial Midwest.

He received some good news Wednesday after The Detroit News and WDIV-TV released a survey that showed Mrs. Clinton with a 43 percent to 38.5 percent lead over Mr. Trump among likely voters. Mr. Obama won the state by 16 points in 2008 and by 9 points in 2012.

The survey found about half of independent voters in Michigan viewed Mr. Trump unfavorably compared with almost seven in 10 who viewed Mrs. Clinton in a negative light.

“Trump puts Michigan in play,” said Saul Anuzis, a former chairman of the Michigan Republican Party. He supported Sen. Ted Cruz of Texas in the primary and now plans to back the party’s nominee.

“Michigan has a strong base of Reagan Democrats and tea party voters that cross party lines,” Mr. Anuzis said. “Being virtually within the margin of error at this point bodes well for Trump.”

In New Jersey, where Mr. Obama won by 15 points in 2008 and 18 points in 2012, Mr. Trump is down by just 4 points in a Monmouth University Poll released Tuesday.

Mr. Murray, director of the Monmouth poll, said Mr. Trump is benefiting from having sewn up his party’s nomination quickly, giving him a head start in consolidating the Republican Party after a divisive primary.

Mrs. Clinton, meanwhile, is still struggling to see off Sen. Bernard Sanders in an increasingly bitter primary race.

Mr. Murray cautioned against reading too much into polls at this point, pointing to Sen. John McCain’s lead over Mr. Obama at this time in 2008. Mr. McCain went on to lose to Mr. Obama by 15 points.