Liverpool and Tottenham have both been heavily linked with a move for the former Real Madrid hitman, who is reportedly unhappy in the French capital.
But Boro have now set their sights on the 23-year-old, according to reports, and are preparing to register their interest.
Boss Aitor Karanka is said to be a huge fan of Jese's talents and is keen to add him to the large Spanish contingent he already has at the club.
Middlesbrough are currently 16th in the Premier League and struggling to score the goals they need to stave off the threat of relegation.