According to the Guardian, Chelsea could offer PSG more than £51m (€60m) for the 30-year-old Spain international, whose reputation has grown further after a prominent role in the club's back-to-back Champions League triumphs.
Ruiz is, however, understood to prefer staying in the French capital, having recently signed a contract extension that runs to 2029, making any deal difficult to complete before the transfer window closes.
He was capped eight times as Spain won this summer's World Cup, adding to a collection of major honours that already included two Champions League winners' medals with PSG.
Alonso's interest in Ruiz comes as Chelsea continue to weigh up midfield reinforcements following Andrey Santos' departure to Manchester United and ongoing fitness concerns over Romeo Lavia. The club is also said to be exploring a separate move for Arsenal's Martin Zubimendi, a player Alonso worked with during his time at Real Sociedad, giving Chelsea two experienced central midfield options under consideration as the window enters its final weeks.
Ruiz made 38 appearances in all competitions for PSG last season, contributing four goals and five assists, though his role fluctuated at times under Luis Enrique, starting only 13 of PSG's 20 Ligue 1 matches he featured in.
Any deal would represent a significant outlay for a player who only recently recommitted his future to Paris, and comes despite Chelsea already having invested heavily in the position with the earlier signings of Enzo Fernandez and Moises Caicedo.