Branthwaite is available for the final seven games of the Premier League season after recovering from hamstring surgery.
He told the Liverpool Echo:“We - as a collective - have made no secret that our goal this season is to get back into a European competition.
"It’s an ambitious one after some tough years but we all believe we have the quality to do so and the target is still there for us as we approach the final part of the season.
“It’s right that we keep looking up now, and keep setting targets that move us forward as a club. It felt like the beginning of a new chapter heading into this season: a new home, our new owners, the manager beginning his first full season since returning.
“We want to keep making progress and, ultimately, deliver success. I’ve been here for six years and I’ve seen a lot in that time but it all feels genuinely positive right now. Long may it continue.
“As a player, you think about winning trophies and to be part of a side that ends Everton’s wait for one would be absolutely massive. This isn’t a team or a club that will be fighting against relegation any more, we want success.
"I feel like the way the club is going, the way the manager is trying to build a team, build an identity, we can definitely achieve that in the next couple of years.”