Citing an unnamed official, the Times said Vice President Mike Pence and acting White House chief of staff Mick Mulvaney had reached out to Webb. It said what it described as a senior Defence Department official confirmed Webb's name had been circulating at the White House.
Jim Mattis stepped down from his job as secretary of defence on Tuesday and President Donald Trump said a day later he had essentially fired Mattis, a retired Marine general whose letter of resignation was seen as a sharp rebuke to the Republican president.
Mattis resigned after Trump's surprise decisions to withdraw all U.S. troops from Syria and half of the 14,000-strong contingent in Afghanistan. Trump has named Patrick Shanahan, a former Boeing Co executive who was Mattis's deputy, as acting defence secretary.