The 23-year-old has impressed on loan at La Liga side Eibar this season, and is owned by Atletico Madrid.
He has scored 17 goals in 33 league games this season, with a further strike coming in the Copa Del Rey.
Both sides face competition from Valencia, while there is also interest from unnamed Bundesliga sides.
AS report that Atletico Madrid will agree to a sale this summer providing they can also strengthen up front, although could insist on a buy-back clause being added to the deal. Baston's current contract at Atletico runs out in 2017.
Both Spurs and Everton need striking reinforcements to add behind star men Harry Kane and Romelu Lukaku, while the latter could need replacing if he forces a move to a club where he can play Champions League football next season.