Clarets boss Sean Dyche was keen on signing the Sweden international, though Mainz's asking price proved too steep.
Mainz refused to budge from demands of meeting his €15m buyout clause - a figure Burnley refused to pay.
"It was close,"said Mainz sports director Rouven Schröder, though refused to confirm it was Burnley who were involved.
It is now likely that Quaison will remain with Mainz this season. His contract expires next summer.