Marney picked up the problem during Sunday's 2-2 Premier League draw at home to West Brom, with subsequent scans revealing the extent of the issue.
The 31-year-old is to seek specialist advice over the coming days with a view to being ready for the start of the 2015-16 campaign.
Having started his career at Tottenham, Marney began to see more first-team action when he joined Hull City in 2006 following a series of loan spells elsewhere.
After four years at the KC Stadium, Marney was snapped up by then Burnley boss Brian Laws in 2010, and has become a key figure during the club's first season back in the top flight.
Marney has started 20 of Burnley's 24 Premier League matches this term.
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