Middlesbrough sit 16th in the Premier League - just four points above the relegation zone.
Manager Aitor Karanka is understood to be seeking deals during the January transfer window, which opened today.
And the Daily Mirror claim Karanka will try to sign Spaniards Deulofeu and Bojan before deadline day.
According to the report, Karanka is an admirer of Everton’s Deulofeu, who has dropped down the Goodison Park pecking order this season.
Deulofeu, 22, has played just 12 games for Everton this term, with his last outing coming in the 3-2 loss at Watford on December 10.
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Middlesbrough reportedly want to sign Deulofeu and Bojan (right)
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Stoke forward Bojan could be tempted to join Boro
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Everton playmaker Gerard Deulofeu has made just 12 appearances for the club this season
As for Bojan, Karanka is said to have asked about his availability as Stoke themselves seek to sign new forwards.
Bojan, 26, joined the Potters in a £3m deal from Barcelona in 2014 and has scored three goals for the club this season, as well as a strike in the Football League Trophy for Stoke U23s.