The Cabinet Office said the election for the vacant seat, planned for 4 May, would now take place on 8 June as part of the nationwide polling day.
MPs will be asked to formally overturn the writ for the poll.
The proposed by-election would have created the anomaly of taking place after Parliament had been dissolved.
The by-election was caused by the death of long-serving Labour MP Sir Gerald Kaufman in February.
Commons Leader David Lidington told MPs there was no statutory provision allowing for the cancellation of a by-election when a general election was in progress.
But he said a precedent was set in a similar situation in 1923, when a by-election writ was regarded as having been superseded.
The Commons vote would enable Manchester's acting returning officer Joanne Roney to cancel the by-election.
A by-election for the Manchester Gorton constituency is set to be cancelled after the House of Commons backed a June general election.