His position has looked increasingly uncertain after Erin Johnson, the chief communications officer, filed an explosive lawsuit last week in New York in which she alleged that Martinez made sexist and racist comments.
She claimed that Martinez had talked about whether he should "rape" a female member of staff "into submission", referred to black people as "monkeys" and complained about Jews.
Johnson, who is seeking unspecified damages, also claims to have video evidence.
Martinez has said there is "absolutely no truth to these outlandish allegations" and WPP said immediately after news broke of Johnson’s complaint that it has "found nothing as yet to substantiate these charges".
WPP has hired a US law firm to conduct "an independent investigation". A source said that there may be "contradictions" between what Johnson and others say happened – particularly at a JWT event in a Miami hotel attended by about 60 staff.
Martinez remains in charge and it is thought WPP is not ready to make any decision until hearing evidence from its lawyers.
One person close to WPP, who knows the Argentine-born Martinez, points out he does not speak perfect English and could be misunderstood.
However, observers believe that he could step aside or be suspended because of disquiet among staff, clients and the wider ad industry.
Lindsey Clay, the president of Wacl, the women’s networking club and former JWT employee, said: "It hasn’t been proven what Martinez said yet. But if it’s true, it’s utterly disgraceful and unacceptable." One person close to JWT said: "Clients will not tolerate this."
Insiders suggest that Stefano Zunino or Toby Hoare, two of JWT’s regional bosses, could run the agency on an interim basis. Another option could be Martinez’s predecessor, the non-executive chairman, Bob Jeffrey.
WPP declined to comment, but a source dismissed suggestions that it is readying a replacement for Martinez.
The claims about Martinez took some in the ad industry by surprise. Doug Quenqua, the editor of Campaign US, wrote about a recent interview in which he thought he heard the JWT boss complain about "too many Jews" living in a suburb of New York but dismissed it, assuming he had misheard.
Saatchi & Saatchi’s New York office posted a Tweet in support of Johnson but later deleted it in a sign of the sensitivity around the row.