United confirmed last Monday that they had made a first offer for the former Arsenal star, which Barca rejected.
But manager David Moyes said at a press conference in Yokohama today that a second offer had been lodged following chief executive Ed Woodward's return to England.
Moyes said: "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."
Woodward had flown back from United pre-season tour ahead of schedule last Wednesday to carry out some urgent transfer business.