Ba, 29, indicated on social networking site Twitter that he had arrived in Istanbul, posting a picture of himself wearing a Besiktas shirt.
The transfer has been declared with the stock exchange in Turkey, but Besiktas have yet to confirm the signing.
Ba is likely to join Besiktas at their pre-season training camp in England once he has joined.
The former Newcastle forward only started 23 of his 51 appearances for the Blues, scoring 14 goals as he found himself third choice behind Samuel Eto'o and Fernando Torres.
And while Eto'o has since been released, Spain international Diego Costa has arrived for £32m and Ba appears to be surplus to requirements.
Besiktas, who are managed by former West Ham and Everton defender Slaven Bilic, are 13-time Turkish champions.
Last season they finished third in Super Lig, behind Istanbul rivals Fenerbahce and Galatasaray.
"Ba is a player we all believe in, we put in a lot of time and effort to complete the transfer," said Besiktas president Fikret Orman. "The transfer is within our budget and we are delighted to have completed the move."
Ba joined the Blues from Premier League rivals Newcastle in January 2013 on a three-and-a-half year deal for an undisclosed fee.
It was thought his contract with the St James' Park club allowed him to speak to other teams if they submitted a £7m bid.
He scored 29 goals in 57 appearances for the Magpies after joining on a free transfer from West Ham United in June 2011.
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