Boro will hope to book their place in the fifth round having reached the last eight in the Capital One Cup, but that would simply represent a bonus to their manager.
Mowbray told the club's official website, www.mfc.co.uk: "Our only ambition is to be a Premier League football club. This
club has a responsibility to the supporters to be competing in the Premier League.
"They were in the Premier League for 11 years on the bounce and this is the fourth season out of it, so it would be wrong of me to say our goal is anything other than to try and achieve promotion.
"If we fall away like we did last year and if we fail, then it hasn't been a successful season.
"Whoever the manager is, whoever the players are, the club has a responsibility to try and take Boro back to the Premier League."