The Spaniard has emerged as favourite for the role, and sources have confirmed to Flashscore that initial exploratory talks have already taken place.
Alonso is understood to be open-minded about the opportunity, although one of his main priorities is gaining clarity over the level of influence and control he would have at Stamford Bridge.
While the process remains in its early stages, Alonso is viewed as one of Chelsea’s leading candidates alongside Cesc Fabregas.
Those in charge at Chelsea recognise the importance of building a clear identity and restoring supporters’ belief in the club’s long-term direction.
Sporting directors Paul Winstanley and Laurence Stewart are spearheading the search for the right appointment at a time when Chelsea’s poor end to the season has increased tensions among supporters, leading to protests against both the ownership and the sporting structure.
Chelsea do have a wider shortlist should moves for Alonso or Fabregas fail to materialise, but for now the focus remains firmly on those two names. They would like a new boss in place before the World Cup begins.
Flashscore reported last week that there is growing concern internally over how effectively Chelsea’s transfer strategy can be executed given the club’s current trajectory.
The prospect of missing out on European football, combined with a decline in the club’s standing within the game, could make it increasingly difficult to attract the calibre of player they ideally want to sign.
As a result, the Blues are open to recruiting more experienced and established players this summer where possible.
Before that, however, they are keen to resolve the managerial situation.
Chelsea still have an opportunity to salvage some credibility from the campaign when they face Manchester City in the FA Cup final at Wembley this weekend.