The Helensburgh RNLI Lifeboat, several police launch boats and rescue craft from boats in the area rushed to the scene when the alarm was raised around 3:30pm on Friday.
The coast guard rescue helicopter and coast guard rescue teams were also at the scene.
The coast guard said there were two people on board the ship when it capsized.
They finished the search around 8 p.m.
Police have cordoned off the area near Custom House Quay in the city of Inverclyde.
A coast guard spokesperson said: “HM Coastguard have been responding to reports of a capsized boat off Custom House Quay in Greenock today, February 24.
“It is believed that the ship had two crew members on board.
“Just before 3:30pm, coastguard rescue teams from Helensburgh and Greenock, a Helensburgh RNLI lifeboat and coastguard helicopter from Prestwick were dispatched to assist and search the area.
“Several boats on the Clyde in the vicinity of the incident also responded, including a MOD police boat.
“Coast Guard participation in the surface search ended at 8 p.m.
The search will resume at 8 a.m. Saturday morning, police confirmed.
A Police Scotland spokesman said: “Police learned that a tug had capsized off Custom House Quay in Greenock at around 3:30pm on February 24.
“Emergency services, including HM Coastguard and RNLI, attended.
“Officers including Police Scotland’s Marine and Diving Unit and Air Support Unit have been conducting searches in the area and these searches will resume on the morning of February 25.
“Investigations are ongoing, assisted by partners, to establish the full circumstances.”