Speculation that Mr. Booker would run against Gov. Chris Christie—a Republican whom he has supported on education policy but not on many social or economic issues—has increased in recent months. Democrats have said the nationally known mayor is one of their best shots against the formidable incumbent…

Although Mr. Booker has the ability to raise millions of dollars from donors across the country, some of those potential backers have expressed reservations about spending the kind of money that would be necessary for a race against Mr. Christie, one of the people said.

While Democrats see the benefit of a Booker gubernatorial candidacy for their own down-ticket races, education advocates expressed interest in a reform-minded politician running for the Senate, the person said.