WASHINGTON — Attorney General William P. Barr dismissed President Trump’s attempts to rebrand the Russia investigation as a criminal plot engineered by former President Barack Obama, saying on Monday that he expected no charges against either Mr. Obama or former Vice President Joseph R. Biden Jr. as a result of an investigation into how their administration handled Russian election interference.

“As long as I’m attorney general, the criminal justice system will not be used for partisan political ends,” Mr. Barr said during a news conference announcing that a Saudi gunman had been in communication with Al Qaeda before a deadly attack last year at a military base in Pensacola, Fla.

Mr. Barr said that John H. Durham, the federal prosecutor investigating how law enforcement and intelligence officials confronted Russia’s operations to meddle in the 2016 election, was examining some aspects of the case as potential crimes but that he was focused on other people, not Mr. Obama or Mr. Biden.

Mr. Trump has in recent days targeted Mr. Obama in public remarks in a way that no modern sitting president has, accusing his predecessor of unspecified crimes under a vague but politically charged catchphrase, “Obamagate.”