As acting attorney general in matters related to the election — because of Attorney General Jeff Sessions' recusal — Rosenstein has oversight over special counsel Robert Mueller, who The Washington Post reported has expanded the Russia inquiry to include a probe of whether Trump obstructed justice.
At an off-camera briefing at the White House on Monday, Spicer was asked about Rosenstein's standing in the administration.
"The president has confidence of everyone who serves for him in this administration," said Spicer.
Asked if that included Rosenstein, Spicer responded, "He serves in this administration."
In testimony before the Senate Appropriations Committee last Tuesday, Rosenstein pledged that Mueller would maintain his independence. Rosenstein added that he would not follow any orders to fire the special counsel unless he believes "those are lawful and appropriate orders."
Earlier that week, Newsmax CEO Chris Ruddy, a close friend of Trump's, claimed that the president was considering ordering Rosenstein to fire Mueller.
In May, ABC News reported that Rosenstein was displeased with the White House portrayal of FBI Director James Comey's firing as originating from him and was on the verge of resigning, according to an administration source.