WASHINGTON — He is fighting two wars, trying to tame a recession and seeking to remake one-sixth of the American economy by overhauling health care. But President Obama is also the world’s best-known Chicagoan. So should he leave his day duties to fly to Denmark to help his adopted hometown win the right to host the Olympics?

Chicago thinks so, or at least some of the city’s leadership does. In the waning days of its bid for the Summer Olympics of 2016, the city is pressuring its favorite son to put it over the top by making an appearance at the International Olympic Committee meeting in Copenhagen next month. It does not hurt that one of the city’s top boosters is Valerie Jarrett, the president’s close friend and White House adviser.

Aides said Thursday that no final decision had been made, but that they doubted Obama would make the trip. It is no easy choice. If he does not go, he risks the consternation of his political ally Mayor Richard M. Daley. If he does go and Chicago loses, he could be embarrassed by not bringing home the gold. Even if he were to help Chicago win, he might still be criticized for devoting time to such a venture at a moment of great national challenges.

Since 2005, when Tony Blair, Britain’s prime minister at the time, went to Singapore to lobby successfully for London’s bid to host the 2012 Games, it has become more common for national leaders to personally attend. In 2007, Vladimir Putin, then Russia’s president, followed suit by going to Guatemala, where he won the right to host the 2014 Winter Games in Sochi.

This year, the heads of state representing Chicago’s three rivals — Rio de Janeiro, Madrid and Tokyo — are expected to go to Copenhagen for the Oct. 2 vote by the I.O.C. Chicago, once viewed as the frontrunner, now appears to be lagging, according to Olympics watchers, and Obama could be the deciding factor.

“Bringing Obama with the bid team would obviously have a positive impact,” said Ric Munoz, a Chicago alderman. “President Obama being as well liked as he is, it’s kind of like bringing Michael Jordan or Oprah Winfrey. It’s that star power.”

Larry Probst, the United States Olympic Committee chairman, said Thursday that his organization had been in touch with the White House through “letters and multiple phone calls” regarding Obama’s potential trip.

“I think we have all very clearly communicated to the White House that we think President Obama’s presence will greatly assist the bid,” Probst said at a meeting of Olympic athletes in Chicago.

The catch is that the results of this presidential trip, unlike many others, cannot be negotiated in advance.

“You don’t know what the vote is going to be and it’s not predictable,” said Kevin Sullivan, a White House aide to former President George W. Bush, who now provides strategic communications advice to sports teams. “But I don’t think it would reflect unfavorably on the White House if the president went and it went to one of the other cities.”

Critics, though, said Obama should not waste precious time, especially while facing critical decisions on issues like health care and the war in Afghanistan. “There’s many other things at home he should be concentrating on instead of getting Chicago the Olympics,” said Bob Quellos, co-founder of No Games Chicago, a group opposing the city’s bid.

Obama has taped four video messages for the city, will host Olympic athletes at the White House next week and created a White House office on the Olympics overseen by Jarrett, his senior adviser, who was vice chairwoman of the Chicago 2016 bid group. The White House has said that she will go to Copenhagen for the meeting.

Juliet Macur contributed reporting from Chicago.