The 26-year-old striker is already in Glasgow for a medical ahead of a transfer from the Amsterdam club.
Boerrigter can be added to Celtic's squad for the Champions League third qualifying round ties with Elfsborg, provided his registration is complete by 11pm on Tuesday.
According to the reports in the Dutch press Boerrigter's agent, Soren Lerby, claims the player will sign a three-year contract with the Scottish champions.
Boerrigter is currently in his second spell with Ajax, having risen through the youth ranks in Amsterdam.
His compatriot Virgil van Dijk has already joined Celtic this summer, with the defender arriving from FC Groningen.