The daily tabloid De Telegraaf said the Amsterdam powerhouse and the London club had agreed a fee believed to be around 45 million euros ($50 million) for the 26-year-old Dutch-born Moroccan.
"Ajax's management reached an oral agreement over the past 24 hours with the top London club for the Moroccan attacker to make the switch in the summer," the newspaper said.
Chelsea manager Frank Lampard believed Ziyech "is a top player and such an essential reinforcement to his 11 that he gave the club instructions to have a deal done before the summer," De Telegraaf said.
Clubs can agree and sign transfers outside the English Premier League's January transfer window, but players may not be registered with new clubs until it opens again in the summer.
Ajax declined to comment on the report.
"As always we can't comment on rumours. Any news will be reported through the official Ajax channels," an Ajax spokesman told AFP.
Ajax is known for selling home-grown talent to richer European clubs. After last season, they sold Matthijs de Ligt to Juventus and Frenkie de Jong to Barcelona.
De Telegraaf said the chance to play Champions League football at Chelsea was "an important condition" for Ziyech, who has scored a total of 47 goals in 157 appearances for Ajax.
Ajax reached the semi-final of the competition last year, but this season they were eliminated in the group stage, in part because they lost at home to Chelsea and later threw away a three-goal lead at Stamford Bridge.
The top two finishers in the Dutch league qualify each year and Ajax are top, 10 points clear of third-place Willem II.
Chelsea are fourth in the Premier League, which would earn a Champions League spot, but their poor form has allowed a chasing pack of five clubs to close in.
They are still in this season's Champions League and could qualify for next season by winning it.
Until Ziyech formally signs a contract "the door was still slightly ajar for other top clubs", but Ajax did not want a deal with any other club, De Telegraaf said.
Ziyech joined Ajax in 2016 from fellow Eredivisie club FC Twente for a fee of 11 million euros.
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