Republican senators effectively blocked President Barack Obama's ability to nominate a director for the new consumer financial-protection agency, saying they won't confirm anyone for the post—regardless of party affiliation—unless the agency is restructured.

The move makes it likely Mr. Obama will resort to a so-called recess appointment, tapping someone to run the Consumer Financial Protection Bureau when Congress is out of session. The agency needs a confirmed director in place by July 21 or, according to the legislation...