The scene saw vicar Billy Mayhew, played by Daniel Brocklebank, taking heroin in a church alongside his brother Lee, played by Richard Crehan.
After taking the Class A drug, Lee tells Billy: "This must be what heaven feels like."
It aired before the 9pm watershed during the second episode of Monday's double bill.
A spokeswoman for Ofcom said: "We are assessing these complaints under our broadcasting rules, before deciding whether or not to investigate."
An ITV spokeswoman said: "Coronation Street has a strong tradition of tackling serious contemporary issues in a responsible way.
"In upcoming episodes we will be showing the very negative side of drug taking as Billy's life starts to fall apart."
The spokeswoman added that the storyline was researched thoroughly with the help of the team at Addaction.
Many fans of the ITV soap took to Twitter to complain about the depiction of drug use shown.
One wrote that it was "disgusting" to air before 9pm, and that "someone needs to be sacked".
Another said: "@itvcorrie I used to enjoy watching Coronation Street with my teenage daughter - not anymore.
"What are the script writers trying to achieve? Disgraceful heroin scene."
One added: "Not sure the message that sends out... 'what heaven feels like' @itvcorrie #CoronationStreet."
Viewers have seen Billy descend into a downward spiral of drug addiction in recent episodes.
The latest scene was prompted when he asked Lee for methadone, which he has been using for pain relief, but instead was given heroin.
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