The meeting lasted five hours, according to local media.
Arias was chosen to succeed Sevilla's renowned sporting director Ramon Rodriguez 'Monchi', who left last April to join AS Roma after 17 years in the role and who is widely credited for the club's success over the last 12 years.
"The board has decided that the sporting director will not remain at the club next year. Oscar Arias will remain at the club for now but will work in other areas and will not be the person who plans the team for next season," Castro said.
Montella succeeded Eduardo Berizzo who was sacked in December. The Italian led Sevilla to the Champions League quarter-finals for the first time, with a 2-0 aggregate win over Manchester United, as well as to the King's Cup final.
He has presided over a poor run of results in the league, however, losing seven of his 16 games, drawing four and winning five, which has seen Sevilla slip to seventh in the Liga standings. They are without a win in six league games.
Sevilla have 48 points, level with eighth-placed Getafe and a point ahead of Girona in ninth, albeit with a game in hand. They need a top-seven finish to qualify for the Europa League.
"We're going to finish the season and then we will decide," Castro said, referring to Montella's future at the club.
"On Sunday when we got back from Madrid we met with him. He feels strong and he's confident he can take the team back into Europe. He transmitted confidence in us and we feel the same confidence in him."
(Reporting by Richard Martin; editing by Ken Ferris)
