He was loaned out to West Bromwich Albion in January but failed to make an impact at the Hawthorns outfit, failing to score in six appearances for the relegated club.
Local media reports linked Sturridge with a move away from Liverpool but manager Klopp said this week that the striker still has a future at Anfield.
"I see myself staying at Liverpool, hopefully being part of the team week in, week out," Sturridge told the Daily Mirror.
"Pre-season is going well, so for me it's keeping my head down and building on that. I'm excited about the season ahead and it feels great to be back."
Sturridge, who was left out of Gareth Southgate's England team for the World Cup, has played in all three of Liverpool's pre-season friendlies so far.
Liverpool continue their pre-season fixtures against Blackburn Rovers on Thursday before heading to the United States to face Borussia Dortmund, league rivals Manchester City and Manchester United in the International Champions Cup.
(Reporting by Aditi Prakash in Bengaluru; editing by Sudipto Ganguly)
