The Brazilian-born forward has emerged as one of Europe’s top goal-getters this season, netting a staggering 23 times in 22 outings for the high-flying La Liga side.
His razor-sharp form has led to links with Arsenal and Manchester United, but Mourinho is keen to beat his rivals to the 25-year-old.
Costa has committed his future to the Madrid side for the remainder of the season, however a summer switch could be a possibility.
The powerful frontman has a £32m release clause in his contract, which is a sum that ‘The Special One’ could be willing to trigger to solve his striker issues.
Costa is represented by Mourinho’s agent Jorge Mendes, which could smooth the process of landing him for the Londoners.
The hitman is keen to stay in Spain until at least the end of the current campaign after pledging his international future to La Roja thanks to growing frustrated at a lack of recognition from his birth nation.
Chelsea are also looking at Wayne Rooney, but United’s reluctance to discuss a deal may block a switch.