The two individuals, who have not been named, will now self-isolate in their homes as part of quarantine conditions and will not travel.
Atletico confirmed the players, staff and other members of the travelling party had undergone testing at a medical centre in Majadahonda on Saturday.
It was then revealed on Sunday that two tests had been returned positive, with the club communicating their findings to Uefa and the Portuguese and Spanish football authorities as required.
New tests will now be required to establish whether the virus has spread to other members of Atletico's contingent and training has been postponed to Monday as they await the new results.
The Spanish club will now discuss a revised schedule with Uefa ahead of their last-eight clash in Lisbon, which is due to take place on August 13.
A club statement, translated from Spanish, read: "Yesterday, Saturday August 8, all members of the first team and the club's expedition to Lisbon took PCR tests in the Sports City at Majadahonda as required by the UEFA protocol to participate in the quarter-finals of the Champions League.
"Among the results known today there have been two positives, both isolated in their respective homes, and which were immediately communicated to the Spanish and Portuguese health authorities, to UEFA, to the Spanish Football Federation, to the Portuguese Federation and to the National Sports Council.
"In addition, the corresponding protocol of action planned for these cases has been activated, which requires the implementation of new PCR tests for the first team and members of the expedition to Lisbon and to the close contacts of the positive cases, and which motivates changes in the training schedules, in the structure and development of the trip and the accommodation in the Portuguese capital.
"The club will coordinate the new schedules with UEFA and as soon as there is a new plan it will be made public. It also calls for maximum respect for the identity of the two positive cases."