The 28-year-old is out of contract this summer and rejected Bayern’s offer of a new five-year deal worth around £250,000-a-week including bonuses.
Madrid have been in talks with Alaba since last month and are thought to be close to securing Alaba’s signature on a four-year contract worth almost £400,000-a-week.
Other interested clubs have been told they will need to match that figure to enter the running but Chelsea have so far proved reluctant to break their current wage structure.
N’Golo Kante is the club’s top earner on £300,000-a-week. Liverpool were also thought to have expressed an interest but did not pursue an agreement for the same reason.
Chelsea are keen to strengthen their backline this summer having also scouted RB Leipzig‘s Dayot Upamecano, who has a £38 million release clause, with new boss Thomas Tuchel ready to overhaul his options at the back.
Chelsea are open to taking up an option to extend Thiago Silva’s contract by a further year but sources have told ESPN that he is yet to determine whether he sees Antonio Rudiger, Andreas Christensen and Fikayo Tomori, now on loan at AC Milan, as part of his plans for next season.
Alaba turns 29 in June and can command a huge contract after a glittering 11-year career at Bayern which includes over 350 first-team appearances, nine Bundesliga titles and two Champions League wins.