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.
Manchester United eyeing surprise move for AC Milan star for whom tension in Italy is growing: report
‘Even though Kevin Keegan was my boyhood hero, we never really saw eye-to-eye. That was an underwhelming time for me – I felt on the outside’: Newcastle cult hero reveals disappointment over how things went under King Kev
