"We spent four days arguing about those records," he said. "I was explaining to him why it wasn't right and I had an idea for making this album called Magna Carta Holy Grail."
However, the rapper explained that the argument was eventually contained.
"There was some pushing at one point but not between us, just everyone else got a little excited," he said.
Jay-Z and West began working together around 10 years ago with West producing four of Jay-Z's albums and Jay-Z making appearances on three West records.
The 43-year-old rapper added that he also had a "fall out" with Timbaland when he was producing his "Blueprint 3" album as "a couple of songs were getting leaked that were his songs and that whole thing just ruined the process."
The "Holy Grail" singer explained that his experience with Timbaland was "almost the last straw," but added that he had seen a new side of him when he came back.
"He has this maturity and this growth and I was like, 'Oh OK, this is going to work out'," Jay-Z said.
Regarding the idea of working alongside wife Beyonce, Jay-Z said he wouldn't be opposed to it just as long as it wasn't a "gimmick."
"[The idea is] always floating out there," he said. "We just don't want to make it a gimmick. It has to be the real thing and it's slowly making sense as far as a tour, but not an album."
During the radio interview with Lowe, which will air in four parts on Monday, Tuesday, Wednesday and Thursday form 7p.m. on Radio 1, Jay-Z opened up about not wanting to be a politician, the recording of his latest album and being a dad.