The 25-year-old Ghana striker arrived on Wearside last summer initially as a foil for England counterpart Darren Bent, but ultimately found himself taking a leading role after the former Tottenham frontman headed for Aston Villa in January.
Gyan's return of 11 goals in his first campaign was creditable enough as the Black Cats secured a 10th-place finish, and he will embark upon the new season at Liverpool confident there is more to come from him.
He said: "It's a challenge, it's a challenge to me. I came here and I have proved to everybody what I am capable of.
"I think everybody will be expecting a lot, so I have to be ready to make sure I deliver the goals for the fans and for the club.
"I did quite well last season even though I was having problems with injuries, so I will make sure I take good care of myself and make sure I score a couple of goals."
Manager Steve Bruce is in little doubt that English football is yet to see the best of Gyan, and is relishing the prospect of seeing him go through his paces with a full pre-season campaign under his belt.
The African found himself at the centre of speculation that he could be heading away from the north-east during the summer, but the manager never had any intention of allowing him to leave and, having lost loan signing Danny Welbeck and Fraizer Campbell to another long-term injury, he has drafted in Connor Wickham and Ji Dong-won to provide him with support.
Bruce said: "I don't know where the speculation comes from, I really, really don't. I don't know whether it's coming from agents who are meddling or whatever, you never know.
"But certainly as far as I am concerned, I am expecting big things from Asa.
"He has now had a season under his belt, he knows what the Premier League is, he knows we don't have a break at Christmas, so I hope he can have a big, big season.
"The most important thing for him was to get through pre-season. We had six or seven weeks where he hurt his hamstring towards the end of last season, but he is up and running.
"He has had a decent pre-season and I am sure he will be ready for the challenge of Liverpool - if he is selected."
Bruce has had a busy close season during which he has recruited 10 new men, although he insists he will not make wholesale changes.
Opposite number Kenny Dalglish has spent even more heavily, principally on Sunderland midfielder Jordan Henderson and Villa winger Stewart Downing, although Bruce believes former club Manchester United will still be the team to beat in the race for the title.
He said: "Anybody who finishes above Manchester United will win the league. They are the ones.
"They won the league, they have spent wisely again - and I don't think they are finished. They are the ones to beat.
"But Liverpool have certainly made great strides in six months and with Kenny back at the helm, it has rejuvenated them.
"They were in the papers for all the wrong reasons, which is not like Liverpool.
"Over the years, it has been a closed shop there and you only see Liverpool on the back pages.
"We have seen it too often in turmoil, which is not like Liverpool. It has a great history, a great tradition and with Kenny in charge, then they have got every chance."