Inouye, a Democrat, was first elected to the Senate in 1962 and served longer than anyone except Robert Byrd, D-W.Va. He was elected to the House when Hawaii first became a state.
"Dan Inouye was an institution and he deserves to spend at least another day in this beautiful building in which he dedicated his life to represent the 50th state from the first day Hawaii was admitted into the union," Reid said.
When Inouye's coffin arrived at the Capitol, it was placed on a catafalque used in 1865 for President Abraham Lincoln.
The Rotunda was to be open until 8 p.m. to the public. Inouye's funeral is scheduled for Friday morning at the National Cathedral in Washington, The Washington Post said.
Inouye served in the Army from 1942 until 1947, losing an arm during the Italian campaign in 1945. President Bill Clinton awarded him the Medal of Honor.
He is to be buried Sunday in the Punchbowl military cemetery in Hawaii.
Notable deaths of 2014 [PHOTOS]