The Sun and Daily Mail report that the superstar striker of the 1980s - who notched 193 goals for the Blues between 1983 and 1992 - was arrested after a tip-off and taken to a police station in Dunstable for questioning and fingerprinting.
A woman was also arrested with Dixon. Both have been bailed until June 18 pending further enquiries.
A Police spokesman said: "A warrant was executed at an address in Dunstable on May 2. A man and a woman were arrested on suspicion of conspiracy to supply Class A drugs. They have been released on bail."
He also won eight England caps, scoring four goals, though the presence in the squad of Gary Lineker and Peter Beardsley severely hampered his opportunities on the international stage.
Though he has dabbled in football management, Dixon's main employment these days is as a pundit, and he is a regular contributor to Chelsea TV.
Neither Dixon nor Chelsea have commented on the matter.
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