Nishimura said on Sunday that "if vaccination moves ahead at the current pace, it will surpass 60 percent by the end of this month", reports Xinhua news agency.
Although the vaccination rollout started late in Japan compared with other developed economies, the government has put efforts on accelerating the progress with goals to complete the full vaccination of all people who are eligible and willing to receive the shots by early November, he added.
Nishimura also said that if the vaccination rate went up to 80 per cent, it would have a considerable impact on Covid-19 infections.
The vaccination program in Japan first began with healthcare workers in February, then expanded to those aged 65 or older from April, finally moved to people under 65, who could receive shots in some municipalities and at their places of work.
As Japan's medical system was still under strain due to the highly contagious Delta variant, the government decided to extend the state of emergency in Tokyo and 18 prefectures from Monday.
The number of daily Covid-19 infections confirmed in Japan has been decreasing in recent days.
With the expectation of increasing fully vaccinated people, the government has devised plans for a November easing of recommendations against travelling and large events.
The number of daily Covid-19 infections confirmed in Tokyo was 1,067 on Sunday, dropping 786 from a week ago, according to the Metropolitan government.
The capital city's daily tally has declined week-on-week for 21 days in the row.
Daily infections totalled around 7,200 on Sunday across the country, and prefectures in western Japan have reported fewer cases than the previous week for 11 straight days.