For weeks leading up to his announcement on Wednesday, Mr. Biden also repeatedly told advisers and potential campaign staff members that he did not believe Mrs. Clinton could defeat the Republican candidate. He watched as she played up her relationship with Mr. Obama, especially when speaking to black crowds in South Carolina and elsewhere, and argued that if anyone should take advantage of the sitting president’s record and high approval rating among Democratic primary voters, it should be him.

The vice president viewed Mrs. Clinton as “this book-smart student who succeeded by dint of grunt work but not by dazzling brilliance,” saw himself as the more “intuitive politician and the intuitive leader and policy maker,” said one Democrat who has spoken to people advising Mr. Biden.

In the last 48 hours, Mr. Biden wore his growing frustration on his sleeve, offering a thinly veiled warning to Mrs. Clinton that Democrats would be “making a tragic mistake if we walk away or attempt to undo the Obama legacy.” In recent days, Mrs. Clinton has announced her opposition to the president’s trade deal with Asia, among other policies.

As colleagues in the Senate, Mr. Biden and Mrs. Clinton were friendly. He was one of the first people Mrs. Clinton met with after her concession speech when she dropped out of the 2008 primary. After Mr. Obama made his vice-presidential choice known, Mr. Biden told a crowd in Nashua, N.H., that Mrs. Clinton “is as qualified or more qualified than I am.” He added, “And, quite frankly, she may have been a better pick than me.”