HONOLULU — Among the first announcements President Barack Obama will make after returning from his Hawaiian vacation is his choice for his top economic adviser.

The person who replaces Lawrence Summers, the outgoing director of the National Economic Council, will have a guiding hand in nearly every economic decision the Obama administration makes, and the president’s choice is being closely watched for signs of where he wants to take his economic agenda in the second half of his term.

Will he tap the business world with a figure such as Roger Altman, an investment banker and Clinton administration alumnus who might carry too much baggage from his association with Wall Street?

Will he turn to academia instead, calling on a scholar like Yale President Richard Levin?

Or will he go with deeply experienced insiders such as deficit hawk Gene Sperling at the Treasury Department or Jason Furman, the council’s deputy director?

With the unemployment rate at 9.8 percent, the private sector struggling to maintain steady growth and the public ranking the economy as the top concern, Obama’s handling of the issue is certain to play a central role in his reelection bid.

The selection process for the council post has dragged on for months. Summers announced his resignation in September, and many in the administration knew well before then that he planned to return to Harvard University after serving two years at the White House.

Obama spokesman Robert Gibbs said he expects Obama to make an announcement in early January, and blamed any delay on the frenzied legislative session that consumed the White House through the end of the year.