Koeman was sacked by the Toffees on Monday after a dismal run of form that has seen the Merseyside club slip into the Premier League relegation zone, while qualification from the Europa League group stage looks unlikely.
Names such as Sean Dyche, Phil Neville and Ryan Giggs have been suggested to replace former Southampton boss Koeman, but Bellew has other ideas.
Boxer Bellew, a well-known Everton fan, wrote in his Metro column that the Toffees should splash out £250,000 a week on Simeone and offer the Argentinian manager a huge £300million war chest of owner Farhad Moshiri's money.
"Why not go and break the bank and get Diego Simeone? Offer him £250,000 a week and give him a £300million war chest," Bellew said. "Then let’s see how serious we really are. Simeone has completed his process at Atletico Madrid.
"He needs a new project and what’s better than Everton in the Premier League? Big fan base, a team with money. I’m not too sure who is available at the moment, but I’m desperate to see what happens now.
"I want success at Everton football club."
Simeone, 47, has gained major success at Atletico during his six-year spell as manager, most notably seeing off competition from Real Madrid and Barcelona to hold aloft the 2013-14 La Liga trophy.
Two Champions League finals and a Europa League winners' medal shows just what he is capable of, but it remains to be seen if Everton can lure him to Merseyside with his contract set to expire in June 2020.