Klopp shockingly announced in January that he will leave Liverpool at the end of the season after nine hugely successful years at Anfield.
Alonso was quick to become a heavy favourite to replace him but he has since reaffirmed his commitment to Bayer Leverkusen.
Following the news, Deeney suggested Wolves head coach O’Neil could be the man to kick-start the post-Klopp era at Liverpool.
“They’ve got a culture now,” he told Majestic on The Final Word on talkSPORT. “We heard an Arsenal caller earlier talking about how the club had been away for seven or eight years.
“But now they’re here to stay because [manager Mikel] Arteta has had to build that. I think it’s already there. You’re inheriting people that are used to winning, they’re all internationals, they’re used to winning.
The two then highlighted David Moyes’ struggle when he replaced Sir Alex Ferguson at Manchester United in 2013, a season after guiding them to Premier League glory.
“I’m going to throw a name at you – I think [Liverpool] need a coach next year more than a manager, so I’m going to say Gary O’Neil.”
Since taking over in August, O’Neil has impressively led Wolves into the top half of the Premier League table and three points off the European spots.