Chiefs, who won the Premiership and the Nedbank Cup last term, recently bowed out of the MTN8 semi-finals at the hands of fierce rivals Orlando Pirates.
Stuart Baxter's men are next up against top tier rookies Polokwane City in the last-16 of the TKO on Saturday night, with Nkosi well aware of the Amakhosi faithful's expectations.
"We are not under any kind of pressure but it is our ambition to win every trophy that we compete in because we want to win as much as possible," Nkosi says.
"It is also our ambition to try and win more than we did last season. Last season we lost out on the MTN8 and the Telkom Knockout but our season still went on to finish well.
"Obviously we will put more effort in trying to make up for last season. So it is really important for us to actually do well in this competition because we have already lost out on the other cup that we lost last season."
Meanwhile, the 32-year-old is hungry for game-time after a calf injury hampered his progress early into the campaign.
However, he knows there is plenty of competition at Naturena.
“I am feeling better and getting stronger in terms of match fitness," he says.
"I have tried to make up for lost time by working hard in a bid to catch with the rest of my team-mates in terms of match fitness.
"I have stated before that by coming to Chiefs I always knew that there are a lot of quality players which presents a different challenge.
"However, for me, what I focus on is my personal ambition because I feel that I would still do better even if I was the only player available at the club.
"I will always push myself to the limit with or without competition,” concluded the former Arminia Bielefeld man.
