Current and former Ted Baker staff have accused the fashion brand's founder of instigating a "forced hugging" policy that is part of a culture that "leaves harassment unchallenged".
A petition has been launched on campaign website Organise accusing CEO Ray Kelvin of inappropriate behaviour.
"There are a lot of really positive things about working at Ted Baker but they’re often overshadowed by the ‘hugging’ and inappropriate touching and comments," the petition says.
A Ted Baker spokesperson said that Mr Kelvin "greets many people he meets with a hug" but they are "absolutely not insisted upon".
The company said the claims are being taken "very seriously" and a "thorough independent investigation" will be carried out.
Mr Kelvin founded Ted Baker in 1988 and is worth £552 million, according to the Sunday Times Rich List.
Among the allegations are that the CEO has asked "young female members of staff to sit on his knee, cuddle him, or let him massage their ears".
The petition also claims Mr Kelvin "regularly makes sexual innuendos at staff".
Those behind the petition say when they have reported issues to HR, they are told: "That's just what he's like."
Addressing Ted Baker's board of directors, the petition says: "Put an end to forced 'hugging' by the CEO. It is part of a culture that leaves harassment unchallenged.
"Please set up a way of reporting harassment to an independent, external body - HR has done nothing with the reports of harassment to date.
"Directors who abuse their power should be held to account. Harassment at Ted Baker is well documented but wilfully ignored by those in charge."
Ted Baker said it has "always placed great importance on the company’s culture" and it was vital that all staff feel "valued and respected at work".
The company said in a statement: "Having very recently been made aware of the petition, we are now in dialogue with Organise to understand the details and nature of the claims.
"We have very clear and robust HR policies in place to provide all employees with a wide range of reporting options for any issues that they might encounter, including a completely confidential and anonymous, 24/7 helpline.
"While the claims made are entirely at odds with the values of our business and those of our CEO, we take them very seriously and will ensure that a thorough independent investigation is carried out. We will then carefully consider the content and recommendations of that report."