The Reds have not played in the competition since the 2009-10 season, but have started this campaign well, winning their two opening Premier League fixtures against Stoke City and Aston Villa.
"Like I've been saying for many years, the target is to get into the top four – that is our objective, it's what the club wants and needs," he told the club's official website.
"That will definitely be the aim. We just take each game as it comes and hopefully we'll be there or thereabouts."
Johnson has been at Anfield since 2009, making 114 league appearances in the process.