Cavaleiro left Fulham in pre-season for Lille.
The winger has so far scored one goal and provided two assists in 22 games for the top-flight club - who are currently fourth in the Ligue 1 table.
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Foot Mercato says Real Betis are showing interest in the 30-year-old.
Cavaleiro is on a contract with Lille that runs through the season, with an option for another year.
It is currently unclear whether the French club will take advantage of that opportunity.