Pritchard was having a scan on Monday.
Neil told the club website: “Immediately, Alex will not be available for the short term.
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“We will get more news on the scan this evening and I don’t think it’s as bad as first feared because the swelling isn’t as bad as we expected, so hopefully that’s a good sign,” he added.
Pritchard has proved a big player since Neil took charge and Sunderland will be hoping he will available soon for some key games in the promotion race.
Sunderland are involved in an intense battle for a play-off spot in League One, having dropped two points at Charlton.