The 21-year-old will now head to London to hold talks over personal terms with the Blues.
Salah, who had been strongly linked with a move to Liverpool, will also undergo a medical examination before the transfer can be confirmed.
The fee involved has not been revealed yet but Salah is believed to be valued at around £11million by the Swiss side.
Salah, who impressed in Basel's two Champions League wins over Chelsea earlier this season, will become Chelsea's third signing of the winter transfer window - following Bertrand Traore and Nemanja Matic - if the deal goes through.
He is another option in the attacking midfield roles, but the pace and width he offers suggests Blues boss Jose Mourinho sees him as an alternative to Manchester United-bound Juan Mata, rather than a direct replacement.
Salah, who has scored 17 goals in 27 appearances for Egypt, was said to be the primary January target for Liverpool manager Brendan Rodgers but he admitted earlier on Thursday that he was unlikely to be bringing anyone in.
"Nothing as yet," was Rodgers' response to questions about transfers. "There have been targets we have looked at but there is nothing at the moment."