Boro announced Dunne's signing on Thursday evening via their official website, along with that of Wycombe goalkeeper Steve Arnold.
While Exeter acknowledge there have been discussions between the 22-year-old and Stevenage, they insist no agreement is in place for his transfer - for which City would be entitled to compensation under the Bosman ruling.
The statement on Exeter's website read: "Stevenage announced that they have signed Exeter City midfielder James Dunne.
"While the club was fully aware that Stevenage were talking to James and there has been contact between the two clubs, no offer or agreement has been received or is in place.
"The former Arsenal man has been offered a deal by Exeter City, but is under 24 and due to the Bosman ruling he is entitled to explore his options elsewhere. It is common knowledge that the Bosman ruling also facilitates a compensation fee should the player decide to leave."