The Sun reported that Chelsea representatives met with Iraola this week, with the Basque manager who leaves Bournemouth at the end of his contract this summer having already rejected two separate approaches from Crystal Palace.
Transfer journalist Nicolo Schira confirmed that Chelsea have also held "positive first talks" with Alonso, with the 44-year-old Spaniard having "given his availability" for a long-term project.
Both managers are represented by the same agency IDUB. Chelsea are aiming to make their appointment before the World Cup begins on June 11.
Alonso left Real Madrid by mutual consent in January having won 24 of his 34 matches in charge. His record at Bayer Leverkusen, where he guided the club to their first Bundesliga title in 2024 without losing a single match, makes him the most coveted available manager in European football. Sky Sports confirmed Chelsea rate his CV as the strongest of all the candidates under consideration.
Iraola's work at Bournemouth — taking the club to a prospective sixth-place finish and the Conference League final — has drawn widespread admiration. Crystal Palace made two formal approaches before Chelsea accelerated their own interest.
Oliver Glasner, Marco Silva, Cesc Fàbregas and Filipe Luís all remain on a broader list. Chelsea hope to finalise their choice ahead of a pre-season that begins in early July. The FA Cup final against Manchester City at Wembley on Saturday represents caretaker Callum McFarlane's final game in temporary charge.
Both Iraola and Alonso are understood to have reservations about the stability of the project given Chelsea's history under BlueCo — four permanent managers dismissed since 2022. Whether the club can offer a sufficiently compelling vision of the future is what will determine which of the two accepts.