Mr. Romney indicates, at the beginning and end of the message, that he is calling on behalf of Mr. Rubio but only urges voters to cast ballots for “a candidate who can defeat Hillary Clinton and who can make us proud,” according to a copy of the message obtained by The New York Times.

The explicit point of the message from Mr. Romney, the 2012 Republican nominee, is less about helping Mr. Rubio than it is about sowing doubts regarding Donald J. Trump.

“If we Republicans were to choose Donald Trump as our nominee, I believe that the prospects for a safe and prosperous future would be greatly diminished — and I’m convinced Donald Trump would lose to Hillary Clinton,” Mr. Romney says in the call.