The 26-year-old’s current deal expires next year and he is yet to reach an agreement on an extension.
The north London club and rivals Manchester United are negotiating with their Dutch counterparts for the Argentina international.
Both English sides believe Martinez is happy to join them and personal terms are not expected to be an issue.
If Ajax get what they want, which is thought to be around £50 million guaranteed, the 24-year-old will have to choose his preferred destination.
Chelsea expect to agree a deal with AC Milan for winger Hakim Ziyech in the next few days.
Ziyech, who has been at Chelsea for two years, isn't a key part of Thomas Tuchel's plans and is set to be allowed to depart.
However, Christian Pulisic is in no rush to leave, after being discussed as a makeweight in the bid for Juventus defender Matthijs de Ligt.
Juventus centre-back De Ligt is one of Chelsea's key targets in this transfer window, with Tuchel eager to bring in reinforcements in that area.
The Italian club would a prefer straight-cash deal and are also said to remain interested in Jorginho – but the midfielder is not currently part of any bid being prepared.
West Ham also admire the 22-year-old, but the Clarets are likely to want more than the £10 million the London clubs would be ready to pay.
The winger has racked up 134 Premier League appearances but is likely to leave following the club's relegation into the Championship.
West Ham have made no progress in the Arnaut Danjuma deal, according to Fabrizio Romano.