"We felt that Manchester United equalled it because it [United's offer] included a loan back option."
Ramsey, who has made just 22 first-team appearances for the Championship side that reached the FA Cup final, is expected to have a medical at Arsenal's Hertfordshire training ground within the next 24 hours.
Ridsdale admitted: "We would have much preferred him to remain as a Cardiff City player or returned here on loan next season had he gone somewhere else.
"The only deal with a loan-back option was with Manchester United but ultimately it is up to the player to decide where he wants to go.
"In an ideal world I would have wished him to go to Manchester United were he to leave us, but we have to be careful as we have relationships with Arsenal and want to do business with them in the future."