Nelsen - who will be in New Zealand on Wednesday to captain the All Whites against Jamaica in an international friendly - was this week in Amman to meet Prince Ali.
"I, and all the players I have talked to, fully support the women being allowed to play football - it's a complete no-brainer for me," said Nelsen after meeting the prince. "We have seen the benefits that women bring to the game and want to encourage their involvement. Barring them because of their culture is the total antithesis to that."
Prince Ali said both he and Nelsen agreed football was a game to be played by everyone. "Talking to players and people on the pitch has been great - they are a vocal group and the feedback we have received has been amazing."
Prince Ali and Cox have helped develop a new headscarf - fastened by velcro and which disengages easily - to deal with the safety issues.
They are meeting with IFAB (FIFA's law-making body) this week to propose a new global set of rules for the wearing of headscarves.
"I am not into politics but we need to allow people from different cultures to participate in the game and we need to tackle this issue now or it could be a catastrophe for those aspiring to play," Prince Ali said.
Cox said the benefits were great. "I really believe these women have a right to play. Football is supposed to be for all - but women can't play because they want to wear something on their heads."
More than 650 million Muslim women could be brought into the game.