The newspaper cited sources as saying the delays had been caused by factory lockdowns and the absence of workers during the coronavirus pandemic.
Back in March, there were reports that Apple could be forced to put off the launch of the 5G version of the iPhone.
And the following month, there was a report that mass iPhone production, which normally occurs just before Apple announces its line-up of devices to be released in September, would be delayed by about a month.
Apart from them, smaller Chinese firms like OPPO and OnePlus, both subsidiaries of the electronics giant BBK Electronics, and South Korea's LG have also launched models featuring the new technology.
The Nikkei Asian Review's sources said that Apple had pushed hard to reduce the delay and it was now unlikely that the release of the iPhone 5G would be delayed into 2021.
While Apple enforced work-at-home orders in March, several of its departments have now returned to work, underlining the big role that engineering and testing, neither of which can be done remotely, play before product release.
But another source told the newspaper that the situation remained fluid and final assembly could be delayed to early October in some cases, with a lot of testing still remaining and designs not yet finalised.
Apple shipped about 191 million iPhones last year, a drop of about 8.5% from 2018, marking a second consecutive year of falls, according to the analyst firm IDC.
The Nikkei Asian Review quoted Jeff Pu, an analyst with GF Securities, as saying that this year's shipments were expected to be about 1.84 million units, a drop of 4% from 2019.
But shipments by Samsung and Huawei, the closest rivals to Apple, are expected to fall by bigger margins: 15% and 19% respectively, with the Huawei drop attributed to the US ban on its use of the official Android operating system and the Samsung decrease to the impact of COVID-19 on demand.