The Bosnia international has largely played the role of deputy to Thibaut Courtois.
Conte confirmed Begovic was wanted by Bournemouth last week.
And Burley, who played spent eight years at Chelsea during his playing days, believes the Blues boss will let the goalkeeper leave providing he can land a replacement.
Craig Burley believes Antonio Conte will sell Asmir Begovic this month
Asmir Begovic is wanted by Bournemouth
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“It’s been a decent fit for Jack Wilshere just to get some game-time,” he told ESPN.
“Antonio Conte has already said there’s been an offer. It’s a difficult one because apparently he’s a very good pro, a very good goalkeeper, and a great back-up to have for Courtois.
“But it’s difficult for a manager to stand in the way of a player who I think is good enough to be playing in somebody else’s first team at a very good level.
“There was talk of West Ham, who have Adrian and Darren Randolph, and I think he’s a better goalkeeper than both.
Asmir Begovic has had to play second fiddle to Thibaut Courtois
“I think Conte will show his sympathetic side and potentially let him go in January as long as he can get some cover for Courtois.”
Conte previously claimed Begovic was an important player for the club.
"I want to say Asmir is an important player on and off the pitch. He knows this," Conte said.
"There is this offer. Me and the club we are evaluating this offer.
"The club and the player know my position that, for me, Asmir is an important player."