Even though both groups describe their proposed schools as "Modern Orthodox", there are differences in ethos, with the Chief Rabbi saying that he could not support Barkai.
Until last week, the Chief Rabbi had expressed reservations about the free-school model because, under current rules, only half of its places can be allocated on the basis of faith.
But he came out in favour of Kavanah after the government's announcement of plans to lift the cap on admissions.
Whereas the Kavanah group opted to operate under the aegis of the Chief Rabbinate, Barkai's backers indicated that "they do not wish to", the Office of the Chief Rabbi explained, "since they expect there will be elements of their ethos which would not meet with his approval and would not be consistent with normative United Synagogue practice".
Rabbi Mirvis believed it would "be both irresponsible and impracticable to try to open more than one new school," the OCR added. He hoped to encourage collaboration so that the community would be "served by one application with broad appeal".
Eve Sacks, Barkai's project co-ordinator, said the group was proud of the college's projected ethos which "has been affirmed by the hundreds of parents we have spoken with during the parlour evenings and consultation process".
The group hoped to work "with community organisations including those who are committed to Jewish education and literacy, even if our ethos is not identical", she said.
Neither the OCR, United Synagogue or Barkai would spell out this week where Barkai's religious policy differed from the US. The group's website, however, indicates it would let girls leyn from the Torah in female prayer groups - a practice Rabbi Mirvis has refused to permit in his synagogues. But the Barkai team, which includes former US minister Rabbi Naftali Brawer, say they have no plans to allow partnership minyans (Orthodox egalitarian services).
Barkai's application to the DfE states that its Modern Orthodox outlook would be combined with a "deep respect for other denominations of Judaism, for other faiths and those of no faith".
According to a source close to the Barkai team, it would be prepared to let non-Orthodox rabbis come and speak on certain issues.
Kavanah, which has published less detail about its religious aspects, says it would offer "a perspective on, and understanding of, the world around us, without preaching religion".
Kavanah was the result of a merger, announced earlier this month, between two groups, Kedem High School and the Hertfordshire Jewish Free School.
It was Kedem that first went public with plans to open a new school in north-west London in early February amid parental concern about increasing competition for places.
While some in the Jewish educational world had questioned the call for a new school, the idea has won greater acceptance with Pajes predicting the need for more than 100 additional secondary places in the area within a few years.
Alternative suggestions, such as expanding the intake of existing Jewish schools in the area, had proved "beyond their means", Rabbi Meyer confirmed.
