Those onboard were initially delayed from being allowed to leave the ship, but were eventually permitted to disembark at about 11:30.
The ship docked in Dundee after a two-week tour of Shetland, Iceland and the Faroe Islands.
Coastguard aircraft from as far afield as Prestwick and Carlisle have been scrambled to assist in the search alongside a “commercial” search and rescue helicopter.
Lifeboat crews from Anstruther and Broughty Ferry and Coastguard rescue teams from Dundee, Arbroath and St Andrews are also involved, with support from nearby vessels.
A spokesperson for Ambassador said it was following “stringent search and rescue protocols” and was working with the coastguard and police.
They added: “Ambassador remains committed to the safety and wellbeing of all our guests and crew.”
Forth Ports, which manages the Port of Dundee, said: “We have been informed of an issue on board the Ambition cruise ship while at sea.
“The vessel is currently berthed in Dundee and we are liaising with the ship and its owner.”
The MS Ambition was previously chartered by the Scottish government to provide temporary housing for Ukrainian refugees on the River Clyde.