The Everton forward recently finalised a move to the Red Devils, who are believed to have paid a fee of £75m, which could rise to £90m with add-ons.
Lukaku, who scored 25 Premier League goals last season, has previously represented Chelsea, West Bromwich Albion and the Toffees, but Giggs believes that the pressure will be intensified when he puts on the red shirt of United.
"He will be under scrutiny that he's never had before at West Brom, or Chelsea, or Everton," Giggs told Sky Sports News.
"The pressure will be on you. So he needs to develop a thick skin, he needs to listen to the coaches, because that's all that counts really, and if he misses a chance, the next one that comes along he needs to put it away because, as a centre-forward at United, you're under scrutiny all the time. I think he will score goals, but you need goals from all over the park.
"I've seen a bit of [Alvaro] Morata; I've seen more of Lukaku. The pluses with Lukaku are obviously that he's played in the Premier League and Jose [Mourinho] has worked with him before, and he's got friends. He's friends with Paul Pogba, and that makes the transition that bit easier."
Lukaku has been handed the number nine shirt following Zlatan Ibrahimovic's departure.