Kit Malthouse was speaking yesterday at a Conservative Party Conference fringe event on home ownership organised by the ConservativeHome website.
Malthouse wrote to local authorities on 26 July to reiterate the government’s ambition to see one million homes built in the region by 2050.
The minister invited councils to submit "ambitious proposals" for growth with a deadline of 14 September.
He told the fringe event yesterday afternoon: "I recently sent a letter out to councils in the Oxford-Cambridge corridor saying bring forward your ideas for new settlements and I think we've had 14 responses, something like that."
Malthouse revealed the level of interest as he claimed that local authorities were now more willing to accept the need for large-scale new settlements, in part due to the availability of government infrastructure funding.
He also said the government's garden towns prospectus, published in August, "has had quite a strong response" from interested councils.
Malthouse added: "The conversations I've had thus far with local authorities don't indicate a paucity of desire to look at some of those schemes, particularly in strong growth areas.
"I get the impression things are changing around this idea. Actually, more and more local authorities are accepting the need for new settlements. They can see the government is willing to put significant infrastructure funding behind some of these projects."
Elsewhere, Malthouse said there was a "moral duty" to help young people access the housing they need.
He also said the government was "doing quite well" in accelerating housing delivery.
Last year's 217,000 net new additions, he said, was about 50 per cent higher than the low point of 144,000 net new additions in 2010 following the crash.
He added: "The most recent figures for planning permissions were just over 350,000 a year. So the volume of units is coming through.
"There are something like 892,000 extant planning permissions out there, about half of which are [under construction] and just under half are yet to start.
"As we get these local authority plans in place, I hope those numbers start to grow again."