Far from the New Hampshire primary, aides to former New York City Mayor Michael Bloomberg eyed the solid victories Tuesday by Bernie Sanders and Donald Trump as potential boosts to a Bloomberg candidacy.

The results, Bloomberg advisers said, were a fresh sign he may have an opening, as Democrats and Republicans embraced candidates the aides said were unlikely to resonate with voters in a general election.

Steve Schmidt, a Republican strategist who served as a senior adviser to Sen. John McCain during his 2008 campaign, said the field of candidates was weak and that the former mayor could win.

“Any combination of them in a general election would open a path for an independent candidate,” said Mr. Schmidt, who isn’t advising Mr. Bloomberg. “There’s a big opening in the middle of the electorate, and Bloomberg is on point in that space.”

Bloomberg aides say they are looking far more closely at the results of Super Tuesday, on March 1, when more states will vote in the primaries and it will be easier to gauge the strength of Hillary Clinton and others in the race.