The 35-year-old extended his stay with the League Two club in June after his short-term contract expired at the end of last season.
He helped Notts win seven of their last 13 games to avoid relegation.
"I believe we're on the cusp of doing something great. Hopefully I can be a big part of that," Ameobi said.
Talking to BBC East Midlands Today, the former Newcastle striker continued: "I feel that we have the quality in this squad to compete with any team in this league.
"Promotion spots, fighting for play-off spots. For me, as a team I think we're good enough to be in those positions."
At Meadow Lane, Ameobi is working under former Newcastle United team-mate Kevin Nolan, who initially brought the former Nigeria international in after taking over as Notts boss in January.
"As soon as I got the phone call, it wasn't really a big decision," Ameobi said.
"I know how he works, what sort of character he is, for me that's a big thing, understanding who you're working for. What he's trying to build is a big part of why I decided to come here. "
Ameobi, who represented Nigeria at the 2014 World Cup having played as a junior for England, said Nolan's demanding pre-season fitness routines have been punishing.
"I think the older you get the harder everything is," he added.
"I think it's my 20th pre-season now. I'm still loving it, even if it's a bit painful at times we know it's all for a reason and hopefully we'll see the results once the games start."
Striker Shola Ameobi says he signed a new deal with Notts County because he has unfinished business with a club capable of challenging for promotion.