The offer, according to reports, is believed to be around £30m.
United have yet to make a marquee signing under Moyes' tenure, but the club have been vocal about their pursuit of the central midfielder.
A first offer made last week - believed to be around £26m - was rejected.
"As I understand it, Ed [Woodward] had a response [to the first offer]. We have made a second offer but Ed is dealing with it rather than me."," Moyes told a press conference in Japan as the club's pre-season tour continues.
"Ed is working hard to make deals happen.
"When you're interested in good players you want to give it every opportunity to materialise and I'll do that."
Woodward, the executive vice-chairman, left the tour early from Sydney to return to Europe and work on deals.
His return was not believed to have anything to do with Wayne Rooney, who United insist is still not for sale.