The Alex, relegated from League One, enquired about the 18-year-old but were turned down immediately.
"I spoke to (United Under-21 coach) Warren Joyce last season about taking him," Collins told BBC Radio Stoke.
"They said he was needed in their 21s and a week later he makes his first-team debut and the rest is history."
Rashford's rise has been meteoric after he scored twice on his United debut in the Europa League against Midtjylland and repeated the feat in his first Premier League game against Arsenal before finishing the season with eight goals in 18 games.
Called into the senior England squad, he scored less than three minutes into his first international appearance against Australia and is now in France for Euro 2016.
"It shows you there are good players in these academies," Collins said.
"There are plenty of kids being produced but opportunity is the key - maybe some of the bigger clubs are missing a trick."
Crewe Alexandra made a bid to sign new Manchester United star Marcus Rashford on loan last season, assistant boss James Collins has revealed.