The Portugal midfielder's deal is running down this season.
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And La Gazzetta dello Sport says Juve management have now acted this week.
Juve have made a pre-contract offer to Bernardo's agents as they seek to bring the 31 year-old to Turn in a Bosman transfer this summer.
Barcelona have also been mentioned as a suitor, though Juve have this week made the first serious pitch to the veteran.