Reuters reported that the premiere party for the show was scheduled in March in Los Angeles but was postponed days before it was set to take place. Apple delayed the party again this week and it is now scheduled for Monday.
"Carpool Karaoke" sees Corden sitting in a car with a celebrity and singing along to songs. Past guests include Bruno Mars, Madonna and former First Lady Michelle Obama.
The delay to the show comes as Apple tries to push forward in the realm of original content, potentially pitting it against the likes of Netflix and Amazon. It's something that Chief Executive Tim Cook has talked up.
"In terms of original content, we have put our toe in the water doing some original content for Apple Music, and that will be rolling out through the year. We are learning from that, and we'll go from there," Cook said on a recent earnings call.
In January, the Wall Street Journal reported that the company is planning to build a significant new business in original TV shows and movies. The technology giant has already announced plans for a reality show called "Planet of the Apps".
"Carpool Karaoke" will be on Apple Music, the firm's music streaming service. Original shows are seen as a way to boost subscriber numbers for the $9.99 a month service. Apple Music has 20 million paying subscribers, but trails behind Spotify which has 50 million, according to official numbers.
Apple analysts have flagged up services such as Apple Music as key for the iPhone maker to continue strong growth in the future