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A departure from Stamford Bridge appears inevitable for Conte as a result of his side’s disappointing Premier League title defence.
A host of big names including former Barcelona chief Luis Enrique have been linked with the gig but Redknapp believes Chelsea owner Roman Abramovich should look towards the south coast for their next manager.
“It looks like Antonio Conte is clinging to the Chelsea job by his fingertips,” the former Liverpool midfielder said.
“If the club make a change this summer, why not go for Eddie?
“The time will come when Howe feels he has outgrown Bournemouth and he is more than capable of becoming Chelsea's first permanent British manager since Glenn Hoddle in 1996.”
If the club make a change this summer, why not go for Eddie?
But Redknapp believes Howe, who has guided Bournemouth from League Two to the Premier League, would prefer to succeed Arsene Wenger at Arsenal.
“Eddie likes to have consistency and continuity,” he said in his Daily Mail column. “For that reason, maybe the Arsenal job would be a better fit once Arsene Wenger decides to leave, but it would be difficult for him to turn down Chelsea should the opportunity arise.
“For any owner, Eddie Howe ticks every box.”
Bournemouth continued their ascent of the Premier League table on Saturday after coming from behind to beat Stoke.
Josh King and Lys Mousset struck late on following Xherdan Shaqiri’s fifth-minute opener.
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Redknapp said: “When Bournemouth fell behind to Stoke, it seemed inevitable that they would battle back.
“Eddie Howe has instilled a real belief and will to win in his side.
“After a rocky start to the season, they are now seven Premier League games unbeaten and seven points above the relegation zone.
“Another top-half finish is not out of the question and it would only highlight what a remarkable job he has done at this club.
“For me, Eddie is ready to manage a top-six side.”
