With their offer of £25.9 million rejected, the Scot said Barcelona have been in contact and an improved deal - reportedly worth £30m plus add-ons - has been submitted
The Premier League champions confirmed last Monday that an opening £25.9 million offer had been made to the Spanish giants, however, vice-president Josep Maria Bartomeu insisted on Friday the 26-year-old is not for sale.
The latest offer is reported to be worth £30m plus add-ons, with the former Arsenal midfielder is contracted to the Blaugrana until 2016.
"My understanding is that [Manchester United's executive vice-chairman Ed Woodward] had a response and he spoke with them. I think we made a second offer but Ed is dealing with that personally, rather than me," Moyes told reporters in Tokyo.
"When you're interested in good players you want to give it every opportunity to materialise and I'll do that. I hope things can continue and move forward.
"At this moment in time I can only tell you that Ed Woodward is working hard trying to make the deals happen. We can only hope that some of them fall into place shortly."Long-serving midfielder Paul Scholes retired at the end of last season, with United's central-midfielder options further depleted following the news Darren Fletcher will miss the start of the new term as he continues to recover from surgery.
United had previously targeted Spain starlet Thiago Alcantara, but the 22-year-old opted to join former Blaugrana manager Pep Guardiola at Bayern Munich.