“I would love to see Michael run — I would love the competition,” Mr. Trump said in a brief interview before a rally of several hundred people who lined up for hours to see him here.

He was responding to a New York Times report that Mr. Bloomberg had asked his advisers to draw up plans for a potential run as an independent candidate under specific circumstances, particularly if Senator Bernie Sanders wins the Democratic primary and the Republicans choose Mr. Trump or Senator Ted Cruz.

“I’d love to see what would happen,” Mr. Trump said. Asked to compare his own business acumen to Mr. Bloomberg’s, Mr. Trump said, “It’s different kinds of business, it’s a very different kind of business, but I would say that if he ran, I’d be very happy about it.”