The England international has damaged his right shoulder and will spend most of pre-season on the sidelines.
Wright is now expected to be on the sidelines for six to eight weeks, and Toffees physio Mick Rathbone admitted he faces a race against time to be fit for the beginning of the new campaign.
"Richard has seen a shoulder specialist in Ipswich today and he was informed that, although the injury doesn't need surgery, he could well be out for between six to eight weeks," said Rathbone.
"Unfortunately, goalkeepers have to dive around and that means he will certainly miss most, if not all, of pre-season training."
The injury may see Everton boss David Moyes step up his hunt for a new goalkeeper although Steve Simonsen and Paul Gerrard will hope to stake a claim for a regular place in the side.
