Contacted by AFP, a spokesman for the tax office refused to comment. saying he could not comment on "specific taxpayer files".
Two football superstars Lionel Messi and Cristiano Ronaldo have already paid substantial fines in Spain for the same offence. A third star, Neymar, is reportedly being investigated in Spain over his taxes.
The tax office suspects the 30-year-old Costa "concealed income" from a sponsorship deal he signed with Adidas shortly before he joined Chelsea from Atletico in 2014. Costa, who has both Brazilian and Spanish citizenship, returned to Atletico in 2018.
Spanish public prosecutors must decide whether they will act on the tax office's suspicions and file a lawsuit against Costa, El Mundo added.
The news comes a week after Costa was slapped with an eight-match ban by the Spanish Football Federation (RFEF) for insulting and grabbing referee Gil Manzano during the first half of Atletico's 2-0 defeat to Barcelona on April 6.
He is just one of several famous footballers who have fallen foul of Spain's tax authorities in recent years.
The player, who joined Italian side Juventus last year, also agreed to pay 18.8 million euros in fines and back taxes to settle the case, according to judicial sources.
Barcelona's Messi paid a two-million-euro fine in 2016 in his own tax wrangle and received a 21-month jail term.
The prison sentence was later reduced to a further fine of 252,000 euros, equivalent to 400 euros per day of the original term.