It emerged this wee that the Premier League champions are willing to make a final €40 million bid after their second approach was rejected.
Moyes has spoken candidly about United's desire to sign the former Arsenal captain in recent weeks, but was unwilling to go into detail when asked about the current state of events.
|10/3||Manchester United are 10/3 with William Hill to sign Cesc Fabregas this summer
The Scot also revealed that Robin van Persie's injury is not as bad as first feared.
The striker was substituted during United's 3-2 defeat against Yokohama F Marinos on Tuesday, which led to criticism of Moyes from prominent fitness expert Raymond Verheijen.
Verheijen branded Moyes a "dinosaur" after the manager admitted to over-training Van Persie in order to boost his fitness, but he will be back in action against Cerezo Osaka on Friday.
"Robin was a precaution the other night," Moyes told reporters. "I expect he and [Danny] Welbeck to both play [on Friday]. Carrick had a tight groin and is doing better but probably won't feature tomorrow. He should resume training over the weekend."
Moyes also confirmed that Nemanja Vidic, Nani and Antonio Valencia are all working towards fitness back in England, while Wayne Rooney is on target to feature in United's friendly against AIK on August 6.
"There was a practice match at the Aon Training Complex the other day - Vida, Nani and Valencia all had playing time in that," he continued.
"Wayne is running and at the level we thought he'd be at. He'll step that up this week."
Moyes also praised the contribution of youngsters Jesse Lingard and Adnan Januzaj, who have impressed despite United having lost two of their three matches this summer.
"Both have done well and adapted well, and made contributions to the team," he added. "It's a tradition at United to try to get young players into the team and both have come through the system.
"Hopefully we can get both some playing time next season but both need to keep their levels up and keep making an impact in training and games."
The United boss believes the humidity in the far east has contributed to his side's defeats, but hopes it will all be worth it come the start of the new season.
"It's very hot and difficult for players from our country who are not used to training and playing in these conditions," he concluded. "I hope we'll benefit from playing in these conditions when we get home. I think we may need to use all of our players tomorrow night to make sure they don't use too much weight or fluid. I'm sure it will be very hot again."