Sources indicate that Chelsea have already made significant progress with the player, with personal terms verbally agreed in principle.
The focus of discussions is now on striking a deal between the two clubs.
Chelsea's first bid was around £13 million and it is thought they are getting close to an agreement worth around £21m.
The Blues have explored alternative options at left-back, but have continued to keep communication open with the Spanish top-tier side, believing there is a chance of finding a compromise.
The player has a release clause of £42m, but it was always anticipated an eventual deal would fall some way short of that figure and now there is growing optimism within Chelsea that progress can be made very soon.
Chavarria is fully on board with the move and has already made his intentions clear to Rayo and he is excited by starting a new chapter in London.
The 28-year-old has been keen to make the switch to Stamford Bridge and does not want to miss the opportunity to work under Chelsea head coach Xabi Alonso.
With the player's desire to join Chelsea not in doubt and personal terms already in place, attention now turns to the formalities of the fine details and terms between the two clubs.
As of Sunday afternoon, Chelsea were hopeful that an agreement is being reached, though, with a deal expected to progress quickly.