United are set on signing the Spaniard to bolster their midfield stocks after the retirement of Paul Scholes last season.
The former Arsenal captain has spent two years at Barcelona, but is not always a first-choice starter, and Moyes has made Fabregas his top target ahead of the new season.
Arsenal are reported to have first option on signing the 26-year-old if he returns to England, but United's intent has been shown with a second bid.
They were said to have made a rejected £25million bid last week, with the new figure estimated to have risen to £30m plus add-ons.
Manchester United's executive vice-chairman Ed Woodward left the club's tour of Australia last week to try and negotiate the purchase, and Moyes revealed a second bid has been officially made on Monday.
"My understanding is that Ed's 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 the press 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."
The Premier League champions will play J.League outfit Yokohama F. Marinos on Tuesday, in a match that will see midfielder Shinji Kagawa play in his home country.
And although Kagawa told Japanese television that he would like to return Dortmund recently, he said for now he is happy in Manchester.
"I am very happy about being a United player and I am looking forward to seeing all my fans and doing my best," Kagawa said.
"I had some problems last season with injuries and am hoping for a better season. But playing as a number 10 doesn’t matter to me. I just want to be in the team."
On Dortmund, Kagawa said: "I felt very proud to see them in the final (of the UEFA Champions League) and I was excited. It was good for them. But I don't feel I made the wrong choice (in moving to United)."
Manchester United have played in two matches on their Asian tour, following up a 1-0 defeat to the Thailand All Stars with a 5-1 win over the A-League All Stars on Saturday.comments