Moyes' successor as Everton boss, Roberto Martinez, was unhappy at United lodging a bid last Friday for star duo Leighton Baines and Marouane Fellaini, claiming it was unusual for United to do business in such a manner.
Alan Pardew was even more annoyed that Arsenal made an offer for Newcastle midfielder Yohan Cabaye just hours before the Magpies' opening Premier League game at Manchester City, triggering the France international's withdrawal.
Even Jose Mourinho's insistence earlier this week that Chelsea would not make another attempt to sign Wayne Rooney until after the Blues had visited United on Monday had the effect of flagging up his continued interest in the 27-year-old.
Yet, given the transfer window closes on September 2, and two lots of league fixtures, plus the Capital One Cup second round have to be played before then, it is hard to work out what an ideal time would be.
"Everybody is allowed to make offers," said Moyes.
"You don't need to take them. You can say no. It is as simple as that. You can do that publicly or privately."
United have so far tried and failed to sign Baines, Fellaini and Cesc Fabregas from Barcelona. They were also interested in Thiago Alcantara before the midfielder opted to join Bayern Munich.
In addition, they are believed to have enquired about marquee purchases Gareth Bale and Cristiano Ronaldo without really coming close to getting either of them.
Now, with Brazilian midfielder Willian on the brink of joining Chelsea, there is a theory United could get Juan Mata in exchange for Rooney.
So far, though, Moyes has been frustrated in his efforts to reinforce his squad.
"There is a need to do it but there's no pressure to do it," said Moyes.
"We won't go chasing for the wrong ones just for the sake of buying.
"We have been after two or three people in the window and we need to be clear in what we do.
"There is always the possibility it might not happen but we are trying."