The attackers opened gunfire as one of the couriers was leaving a branch of Sberbank on Leninsky Prospect with a bag of cash. He was killed immediately. Another of his colleagues then returned fire from the entrance of the bank before being fatally wounded.
"According to confirmed reports, two armed couriers have been killed and one injured," the source said. Three passers-by are also thought to have been wounded. Their condition is unknown.
Police reported that one of the attackers was injured in the shootout and was subsequently detained.
The robbers fled the scene in three cars, and one of them was later found abandoned. A search for its owner is underway.
An interior ministry source said that the armed couriers were carrying 1,000 euros ($1,600).
The attack is the second of its kind in Moscow in slightly more than a month. On June 27, a bank security van was held up in southern Moscow in a brutal attack which left one security guard dead and another in a coma. The robbers fled the scene with a large sum of money but were detained when their vehicle crashed and burst into flames as police pursued them.