The list also includes US firms HOK, Robert A M Stern, Hammond Beeby Rupert Ainge, Beck Architecture, Lake/Flato, Overland Partners and HKS, from Bush's home state of Texas.
The university has reportedly recently acquired 13,500m2 of land for the presidential library to be built under a masterplan by Hillier Associates - soon to become Hillier RMJM.
Presidential libraries are a nationwide network of libraries, part of the US National Archives and Records Administration, which are repositories for preserving records and other historic materials of every US president since Herbert Hoover, who left office in 1933.
It has in recent years become the norm for presidents to choose their home states to build the library documenting their time in the White House.
A shortlist will be announced by the end of July, and a winner will be announced by August.
It is thought the scheme will cost roughly $200 million (£100 million).