Since Alonso essentially dismissed links to other clubs, the main name being discussed is Sporting CP's Ruben Amorim.
However, while there have been a lot of contrasting reports recently, some of the latest claim that talks are breaking down between the two parties and the 39-year-old is not expected to arrive at Anfield this summer.
Other clubs are also showing interest in his services.
According to DaveOCKOP, Wolves manager Gary O'Neil 'has been interviewed' for the soon-to-be vacant manager position at Anfield.
While this does not mean he is a favoured candidate or in pole position to take over from Klopp, he is still being considered by the Liverpool board as they weigh up other options following recent setbacks.
O'Neil took over at Wolves following Julen Lopetegui's exit in August 2023 and he has made a big impact at Molineux.
The West Midlands side has put in some impressive performances this season, including double wins over Tottenham Hotspur and Chelsea.
Before his move to Wolves, O'Neil helped keep Bournemouth afloat in the Premier League but his efforts in the relegation scrap were not enough to confirm his position at the club.
The 40-year-old was sacked after the end of the season but was snapped up by Wolves just months later.
O'Neils efforts with the Wanderers have earned him praise from his fellow Premier League managers.
Could a strong finish this season see him secure a summer move to another club?.