In an emotional farewell video that lasted longer than four minutes, it was unclear whether Silva was choosing not to pen a new contract of his own accord or whether co-owners Todd Boehly and Clearlake Capital took the decision to cut costs at a time when the club must do everything possible to meet the Premier League's Profit and Sustainability regulations.
Nevertheless, while Silva is now 39 years of age and no longer at the peak of his defensive powers, his exit will leave a considerable void in the dressing room in terms of experience.
Boehly and Clearlake have put Chelsea in the peculiar position of having no active senior outfield player in the squad that is above the age of 29.
Head coach Mauricio Pochettino has strongly hinted in recent weeks that he would prefer some experienced heads in his dressing room to help guide his younger crop, and that creates a scenario where an elder statesman could be signed on a free transfer in the summer.
Here, Sports Mole takes a look at four possible options that would fit the bill if Pochettino was allowed to go down that route.
Joel Matip
While he may have only made 201 appearances for Liverpool across an eight-year period, Joel Matip has been one of the club's best pieces of business in recent memory.
Signed on a free transfer from Schalke 04, Matip has gone on to win a Premier League, Champions League, FA Cup, EFL Cup and UEFA Super Cup during his time at Anfield.
Regardless of Jurgen Klopp's departure, the former Cameroon international may have been in line for a new contract had he not suffered a season-ending knee injury in December.
As a result, the expectation is that the 31-year-old will leave on a free transfer. Interested clubs may wait until he has recovered from his latest fitness issue before making a move, but Chelsea could benefit from trying to beat the queue for a player who has 150 Premier League appearances to his name.
Nacho
Staying on the theme of long-serving players, Nacho will end his association with Real Madrid when he departs at the end of his contract in June.
Since the 2010-11 campaign, Nacho has racked up 356 appearances in all competitions, acquiring a remarkable total of 23 pieces of silverware during his time at the Bernabeu.
Nacho may now be 34 years of age but he has still been used on 37 occasions this season, including making 23 starts in La Liga and Champions League fixtures.
Most recently, Nacho played the full 120 minutes as Los Blancos drew 1-1 with Manchester City at the Etihad Stadium, producing a performance that showcased that he is far from over the hill at the highest level of the club game.
Also having the versatility to play in either full-back role, Nacho should arguably be Chelsea's first port of call when it comes to replacing Silva.
Sebastian Coates
As a player who has previously failed to make a sustained impact at either Liverpool and Sunderland, you can understand why Chelsea supporters would have reservations over signing Sebastian Coates.
However, the Uruguay international is now a different player than the one that did not convince at Anfield or the Stadium of Light, having since gone on to establish himself as a Sporting Lisbon legend.
A total of 34 goals have been scored from 303 appearances, while a seventh piece of silverware could soon be added with Sporting closing in on a second Primeira Liga title with Coates in their backline.
Sporting may have a marginally-weaker defensive record compared to Benfica and Porto, yet they have not lost a game in which Coates has featured since January.
Pochettino may need to switch to a back three to make the arrival of Coates work, but there can be no denying that his physical presence would hugely benefit a Chelsea side that lacks height.
Raphael Varane
With Raphael Varane having failed to hit the heights of his Real Madrid career during his time at Manchester United, there will be few Red Devils supporters who will be against his departure in the summer.
Nevertheless, that is not to say that he cannot add value to other clubs in the Premier League. Varane's injury record at Old Trafford has not been great, but this is still a player that has made 93 appearances across three near-completed seasons.
Where the 31-year-old ends up next remains to be seen. A return to Real Madrid cannot be ruled out, especially in line of Nacho's pending exit, while there is every indication that Varane would like to represent Lens again before the end of his career.
Moving to either of those clubs makes more sense than Chelsea, yet the 2018 World Cup winner - who has since retired from international football - may feel like he still has a point to prove in the Premier League.