The deal has been in the pipeline for some time now, but appeared to be in danger of falling through as the two parties attempted to negotiate a fee.
Chelsea reportedly refused to go above £15 million for the full-back with Atletico said to be holding out for much more - and according to the Daily Mail it is the Blues side who blinked first and upped their offer by £5 million.
That has broken the deadlock and Luis looks likely to become a Chelsea player in the coming days. He will join team-mate Diego Costa at Stamford Bridge - the striker has just been confirmed as a Chelsea player after the club issued a statement revealing that they had activated his release clause last month.
The pair will join Cesc Fabregas, who joined Chelsea from Barcelona last month in a deal worth over £30 million.
Chelsea transfer gossip
Jose Mourinho is lining up a swoop for in-demand PSG striker Edinson Cavani, who is also attracting interest from Liverpool and Manchester United (The Express)
Paul Pogba's agent has insisted there is no truth in the rumours linking his client with a move to Chelsea (Daily Mail)
Chelsea have been given a clear run at Sami Khedira after talks with Arsenal stalled over his wage demands (Metro)
Chelsea are interested in signing Raphael Varane, whose future at Real Madrid is in doubt (The Express)
Petr Cech has returned to training amidst speculation he could be sold by Chelsea this summer (The Telegraph)
Chelsea have accepted an offer of £8 million from Besiktas for striker Demba Ba (Eurosport)
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