United announced last Monday that they had made a first offer for the former Arsenal star, and Moyes said at a press conference in Yokohama that a second offer had been lodged.
Executive vice-chairman Ed Woodward returned to the UK ahead of schedule last Wednesday to carry out some urgent transfer business, and Moyes confirmed: "As I understand it, Ed had a response (to the first offer). We have made a second offer but Ed is dealing with it rather than me."
So far, United have not made any new signings this summer, other than Uruguay Under-20 star Guillermo Varela.
And whilst Moyes accepts there may come a time when he has to walk away from the deal, that position has not been reached yet.
"I think a point does come," said Moyes.
"But when you are interested in good players you want to give it every opportunity to materialise.
"I will hope things will continue to move forward.
"At this moment in time I can only tell you Ed Woodward is working hard to try and make the deals happen. We hope some of them will fall into place shortly."