Pressure is mounting on the Premier League to act after a female employee exposed sexist emails Scudamore sent to friends, forcing him to apologise.
"It is increasingly clear steps are needed as a matter of urgency to review governance at the Premier League with a view to improving accountability and tackling head on a culture that demeans women and seems to discourage their involvement in the game's administration," Football Association board member Heather Rabbatts said in a statement on Saturday.
"These challenges go beyond the current situation of chief executive Richard Scudamore, however if the league are to move forward in a positive way then he and they should give serious consideration to his position in the coming days."
A Premier League committee will meet next week and is expected to decide what action, if any, to take against Scudamore.
The 54-year-old had previously backed equality for women in football but a number of female England internationals have called for action to be taken against him.
The FA has said the matter is for the Premier League to deal with even though chairman Greg Dyke described the content of the emails as "totally inappropriate".
Sports Minister Helen Grant said the comments were "completely unacceptable" while Moya Dodd, executive committee member at world football's ruling body FIFA, labelled them as "insulting".
TUESDAY MEETING
Rabbatts chairs the FA's inclusion advisory board (IAB) that is meeting on Tuesday to discuss the case.
"I personally hope progress can be made on all of these fronts so we can feel confident the leaders of football are accountable for their actions and support a culture that genuinely welcomes the participation of women and girls in our national game," she added.
"No one can doubt the tremendous achievements of the Premier League in creating one of the world's great footballing competitions.
"But with that success and the massive public interest it generates comes the obligation to behave responsibly and have in place proper lines of accountability and good governance," said Rabbatts.
"Sadly recent events appear to show these things are currently lacking in the administration of the Premier League and indeed there is growing evidence of a closed culture of sexism, symbolised in the email exchanges which have been made public."
According to media reports Scudamore, who has helped make the Premier League the most lucrative in the world since he took over as chief executive in 1999, earns more than 1.2 million pounds a year.
The competition now has a global TV rights deal worth more than five billion pounds over three years.
(Writing by Pritha Sarkar, editing by Tony Jimenez)
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