Jansson has been a vital part of the Championship club’s team over the last three seasons but looks likely to leave this summer.
BBC reporter Adam Pope tweeted that Brentford were set to sign the centre-back, pending a medical, with a deal likely to be announced next week.
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The figure of £5.5million has been cited by several different reporters.
The 28-year-old has played 120 matches for the club over the last three seasons but his relationship appears to have soured in recent months.
When manager Marcelo Bielsa decided his team should allow Aston Villa to score following a controversial Leeds goal Jansson tried to stop them from scoring despite the instruction from the sideline.