Welbeck completed a £16m move from United to the Gunners at the start of the month and is expected to make his debut during today's Premier League match against Manchester City.
Meulensteen worked with the striker when he was coming through the youth ranks at Old Trafford, and he believes that he will easily fit into the playing style at Arsenal.
"I had him from when he was eight, nine years of age and I think the best of Danny is yet to come," the Daily Star quotes him as saying.
"He is a very versatile player and I'm 100% sure he will do very, very well for Arsenal. Danny is a perfect match for Arsenal. He is such a versatile player."
Welbeck scored twice for England in their 2-0 win over Switzerland in midweek.