According to the Daily Mail, Royal Mail staff managed to discover the identities of some of the people who were part of the mystery shopper scheme, after which they ensured that mail to their addresses was always delivered in good time.
It also alleges that they would ensure mail from post boxes near their houses was not picked up before the advertised final collection time, so that it would arrive at its destination more quickly.
Royal Mail issued a statement in response, saying: "Royal Mail is investigating allegations concerning its mail operations in the Glasgow area. The allegations, received via Postcomm, relate to the possibility of interference in the measurement of Royal Mail's quality of service performance in this area."
It said it was conducting interviews at a local level to assess whether the allegations are accurate.
The mystery shopping scheme is run by Research International, but the Daily Mail says they were unaware of the scam. According to the newspaper, it came to light after a whistleblower raised the alarm.
Postcomm, which regulates the postal industry, said it was conducting a preliminary investigation into serious allegations but that the matter was being treated confidentially.
Royal Mail pays bonuses to staff when it hits delivery targets. It says that 93% of mail posted first class arrives the next day.