The Toffees narrowly avoided relegation in what proved to be a rollercoaster of a 2021-22 campaign.
Now Everton are aiming for a much better season this time around - with Begovic setting an ambitious target.
West Ham clinched seventh last term and will have designs on qualifying for Europe once again.
Meanwhile, Newcastle United finished with plenty of impetus in 11th and are only looking upwards under their Saudi-ownership.
And Aston Villa have also shown their intent by signing Diego Carlos, Philippe Coutinho and Boubacar Kamara this summer - as well as Lucas Digne from Everton in January.
What’s been said
But speaking to talkSPORT, Begovic believes Lampard’s troops can push for a spot in the Europa League or Europa Conference League.
The goalkeeper said: "I go back a year and I've got Everton on the phone and think: What a club, (it's got) everything - tradition, fan base'.
"Now I've seen it, we have to challenge that top seven and eight, those European places.
"We have to be. It's too big of a club, too good of a squad, we've got a big-name manager, all the facilities.
"Everything is in place to be successful. If a few things go our way, we have a good pre-season and get off to a good start, why couldn't we challenge for the top seven?
"I know it's crazy to say a little after the season we have just had but why couldn't we?”
‘We feel what it means to people’
Not many tipped Everton to be in a relegation scrap last term.
But Lampard ended the season with momentum behind him and the fan-base united.
Begovic now believes the Toffees have something to build on.
He added: “Last season wasn't great. We might have lost two or three years of our lives!
“It was terrible. The stress every day and you feel it. We feel what it means to people, especially at our club when no-one expected to be there.
“You understand the consequences and it could have been deep.
“It was a big relief to stay up. It feels like an achievement and it should never have been an achievement.
“But we are looking forward to it. We have a great gaffer, what he did in the final 4-5 months - credit to him.
“We have got a great manager, really good staff and I think we've got a good squad.
“We'll see how the summer works out. We've got some good quality, hopefully have some better luck with injuries and all this kind of stuff.
“We're looking forward to a much better season.”