Frank Lampard has been a busy man in this transfer window with the club embarking on a massive summer spending spree.
So far around £200million has been spent on new players, with Hakim Ziyech, Timo Werner, Ben Chilwell, Thiago Silva, Malang Sarr and Kai Havertz, all arriving.
And the club might not be finished yet with Lampard keen on bringing in another goalkeeper, while Declan Rice is also being linked with a move.
But Anelka expects the west London side to push both teams next season and even thinks they could beat them to the title.
The Frenchman told Stadium Astro: "I think they have a great chance to win the title. It depends on what Liverpool and Manchester City will do as well.
"But he’s got the quality to finish fourth in the league last season and, with the signings, he’s got the quality to win it.
"It depends after in terms of the experience of the players whether they are strong enough to race until the end but the quality is in Chelsea as well.
"So I believe first, second or third. We will see a great Chelsea at the end. You need something more to win the title.
"If you look Liverpool and you look at City, they have young and experienced players.
"I think Frank is clever, he knew what he needed to be able to win and compete against big teams and you need experience.
"When I was playing for Chelsea and Arsenal, we had a young team but also experienced players.
"You still have space for the youngsters to play, they just have to be strong mentally to compete every week.
"This is football, you always have to compete to be able to play. When you’re at a big club, it’s like that."
Chelsea will get their Premier League campaign underway on September 14 when they travel to play Brighton.