The tanker, which is yet to be built, will be called the USNS Harvey Milk, USNI News said.
It cited a notification signed by Secretary of the Navy Ray Mabus.
Milk was one of the first openly gay politicians in the US and was killed a year after winning election to the San Francisco Board of Supervisors.
He served in the US Navy in his youth as a diving officer during the Korean War before being honourably discharged.
Milk was wearing his Navy belt buckle when he was shot dead.
Speaking in 2012, Milk's nephew Stuart Milk said such a move would send "a green light to all the brave men and women who serve our nation: that honesty and authenticity are held up among the highest ideals of of nation's military".
The news delighted San Francisco politician Scott Wiener, who has called for a ship to be named after Milk, saying it was an "incredible day".
"When Harvey Milk served in the military, he couldn't tell anyone who he truly was," he wrote.
"Now our country is telling the men and women who serve, and the entire world, that we honour and support people for who they are."
The idea is not without controversy. After the suggestion was first mooted critics said Milk would have disapproved of lending his name to a Navy ship, given his opposition to the Vietnam War.
Several other civil rights champions are also set to be honoured with ships bearing their names, USNI News said, including former Supreme Court Chief Justice Earl Warren and women's rights activist Sojourner Truth.
The US Navy is set to name a ship after gay rights activist and politician Harvey Milk, according to a report by the US Naval Institute News.