Paul Goodman said: "I think there will be cultural clashes the attitude of Muslims on the licensing of the annual gay pride march won’t necessarily be the same as everyone else, for example.
"National government can’t do everything. Its role in Birmingham or Manchester is essentially limited.
"It’s clear that under the next Conservative government local government will have to do more it will be up to local politicians to place-shape."
Mr Goodman praised the leadership given by his fellow Tories, former Westminster City Council leader Sir Simon Milton and Barnet LBC leader Mike Freer, both of whom, he said, chaired specialist local committees on cohesion.
However, the shadow cohesion minister’s comments at a Policy Exchange Conservative conference fringe event were criticised by the chief executive of the Barrow Cadbury Trust which campaigns for social justice.
Sukhvinder Kaur-Stubbs said the inter-generational divide was often more substantial than the cultural divide.
"I’m not quite as pessimistic as the shadow minister in terms of the potential for cultural clashes," Ms Kaur-Stubbs added.