The agreement is £12.6m (€15m) plus £4.2m (€5m) in add-ons.
The very last details between the two clubs are being filed.
The midfielder is expected to leave for England on Wednesday and then undergo a medical on Thursday.
Casadei is 19 years old and has not made a senior appearance for Inter.
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New Chelsea owners Todd Boehly and Clearlake Capital have spent their first summer in charge at Stamford Bridge conducting a swift overhaul in every department.
The ambitious US bosses have spent a league-topping £176.5m on new signings this summer, primarily on three first-team signings: Marc Cucurella (£63m), Raheem Sterling (£47.5m) and Kalidou Koulibaly (£34m).
Chelsea also want to sign Everton forward Anthony Gordon but have seen their bid of around £40m-£45m rejected.
Conversations between the two clubs have been ongoing for a few days but the London club have now made their interest official. However, it is understood Everton remain uninterested in selling the player and a new contract remains on the table for Gordon.
An improved offer from Chelsea may well test Everton's resolve, though, as Frank Lampard needs to add forward options to his squad before the transfer window ends and a potential deal would raise the finances to make that happen.
Gordon has started both of Everton's Premier League games so far this season, playing as a false No 9 in the absence of Dominic Calvert-Lewin as the striker recovers from a knee injury.
Gordon has made 51 Premier League appearances so far for Everton, scoring four goals.
Sky Sports' Jamie Carragher:
"It's a lot of money [£40m] for a young player who is just coming onto the scene, but I don't see how Everton could accept that. The position they find themselves in right now, lacking attacking players.
"He's the one big hope for the club. The supporters aren't deaf at Everton. They've lost their first two games and they know they're in for a difficult season. They need attacking players to come in so there's no way they can let him go.
Sky Sports' Karen Carney:
"There's been rumours of Christian Pulisic and Hakim Ziyech going so he'd be more of a like-for-like replacement for one of them. He's young, he's an English talent and that's the sort of recruitment they have gone for.
"They want some young and some experienced players. In terms of getting in a recognised centre forward, he's not the type of player or answer that's been spoken about."
Who will be on the move this summer before the transfer window closes at 11pm on September 1?
Keep up-to-date with all the latest transfer news and rumours in our dedicated Transfer Centre blog on Sky Sports' digital platforms. You can also catch up with the ins, outs and analysis on Sky Sports News.
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