Grossman succeeds Richard Holbrooke, who died in December.
Secretary of State Hillary Clinton met with Grossman Monday, and he was introduced to members of Holbrooke's staff, but still must go through a vetting process, The New York Times reported Tuesday.
His official announcement could occur as early as Friday, media outlets reported.
Grossman's last assignment before retiring from foreign service in 2005 was undersecretary for political affairs during the first administration of President George W. Bush, The Washington Post reported. He now heads the Cohen Group, which advises international business clients on overseas enterprises.
Among others considered to succeed Holbrooke were Strobe Talbott, a former deputy secretary of state who heads the Brookings Institution; and Frank G. Wisner, a former ambassador to Egypt recently deployed to urge ousted Egyptian President Hosni Mubarak to declare he would not to run for re-election. Mubarak resigned Friday after more than two weeks of protests by anti-government demonstrators.
Since his death, Holbrooke's special envoy post was filled on an interim basis by his deputy, Frank Ruggiero.