Terry is currently a free agent after leaving Chelsea this summer. He played for the club for 22 years and ended the 2016/17 season by lifting his fifth championship title and 15th major honour.
The 36-year-old has been linked with moves to the MLS and the Chinese Super League, as well as a switch to Championship side Aston Villa. The former England captain has also admitted that he may yet decide to retire from the game.
But Begovic, who himself decided to leave Chelsea for Bournemouth in a £10m deal in May, wants his former skipper to join him on the south coast.
“If he fancies Bournemouth it would be great to have him there,” Begovic said, speaking at the Asmir Begovic Foundation Golf Classic.
“He's a great player and a great guy as well, so from that point of view it would be really, really good.
“He's obviously the best defender ever in the Premier League. He's a legend at Chelsea and he deserves the right to make the right decision for himself and his family.
“I hope he does whatever is right for him and he's happy with the decision, and I wish him all the best for the future.”
Terry is meanwhile keeping his options open and commented shortly before his final Chelsea game that he is happy to wait for the right offer to come along.
“I've still not ruled out Sunday being my last game and retiring from football,” he told Sky Sports.
“It depends, if the right offer comes along I will sit down and consider it with my family - whether that's here, abroad, or wherever that may be.
“Genuinely I haven't made any decisions of yet and I'm evaluating all my options at the moment.”
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