WASHINGTON — Robert S. Mueller III, the special counsel investigating Russia’s interference in the 2016 election, transformed this week what has appeared to be a weakness — his relative lack of independence — into a strength.

Since he was appointed in May by Rod J. Rosenstein, the deputy attorney general, the scope of Mr. Mueller’s power has been intensely debated. President Trump’s allies have portrayed the special counsel as running amok as his focus expanded to the business dealings of Mr. Trump’s associates. The president’s critics have worried that Mr. Mueller is too vulnerable to potential Trump administration interference.

The tension is familiar when the president comes under scrutiny by law enforcement, as the vast powers of the executive branch are at stake. Similar tensions cropped up during the Watergate, Iran-contra, Whitewater and Valerie Plame leak investigations. Now they have spurred the first significant legal fight arising from Mr. Mueller’s work, as Paul Manafort, Mr. Trump’s onetime campaign chairman, battles charges related to payments he received from a pro-Russia government in Ukraine.

“Manafort’s fight over Mueller’s authority goes to a central question in our republic, going all the way back to Plato’s question, ‘Who guards the guardians?’” said Neal Katyal, a former Justice Department official who drafted its special counsel regulations in 1999. “The central debate is how much independence do you want, versus how much accountability for the prosecution. There needs to be a balance between these two competing goals.”