The proposals, which will also transform the club's football academy on the same Trent Vale site in a bid to unearth international stars of the future, were unveiled to the public at the start of the month.
Now a planning application has been submitted to Newcastle Borough Council, and a final decision is expected to be made in July.
The project, occupying seven acres, involves extending an existing academy building, installing an indoor football pitch, four flood-lit synthetic pitches, demountable spectator stands, a running track, salt saturator and floodlighting.
The club's aim is to achive Category 1 status in the Premier League's new Elite Player Performance Plan.
Richard Smith MD of Stoke City Properties Ltd said:
'The club and its owners are firmly committed to achieving Category 1 status which will give us the best possible chance of developing our own players for the first team.'
The scheme will be phased over the next trhee years if given the go-ahead and may lead to over 100 jobs being created.