Mitchell missed defending champions Northampton's defeat at Bath on the opening night of the Premiership last Friday.
Northampton rugby director Phil Dowson said after that match that Mitchell had taken a knock to his neck, with the club awaiting scan results.
Saints continue their Premiership campaign when they host Exeter on Saturday, but Mitchell is unlikely to be in contention.
"He is seeing a specialist again on Thursday, so more update really, then, in terms of timeline and what it looks like," Dowson told reporters during his pre-match press conference.
Mitchell, who has won 18 caps, is England's clear first-choice scrum-half.
Head coach Steve Borthwick will be keeping a close eye on developments with the autumn series opener against New Zealand less than six weeks away.
In-form Bath captain Ben Spencer, who provided bench cover for Mitchell during both summer Tests against the All Blacks, offers quality back-up.
But Mitchell has made the position his own under Borthwick, and any long-term absence would prove a significant blow for club and country.
