The Superfast Cymru contract was set up to connect 96 per cent of businesses and homes excluded from commercial roll-out and the project has so far reached 581,000 premises across Wales, according to BT and the Welsh Government.
Concerns have been raised from residents in rural areas of Mid Wales about the deadline being met, although Superfast Cymru insisted it was "on track".
The latest Welsh Government figures from March showed only 57 per cent of eligible customers in Powys are connected.
In a joint statement, BT and Welsh Government said: "The original Superfast Cymru contract target is anticipated to be completed by BT by end of June 2016, followed by robust Welsh Government testing and verification taking it to summer 2016.
"We are exploring the geographical and technical aspects of extending coverage of superfast broadband to the final few premises in Wales, helping to make sure that no communities are left behind."
A Superfast Cymru spokesman said: "Bringing high-speed broadband to rural areas is a huge engineering task. We will be deploying about 3,000 new fibre broadband cabinets to bring superfast broadband to Wales."
The Welsh Government has part-funded the project, investing £205 million into the initiative but industry experts thinkbroadband.com said they do not believe the deadline can be met.
Andrew Ferguson, editor of thinkbroadband.com, said: "We don't believe 96 per cent can be hit this month at all. Current rates suggest the end of 2016."
With regard to the Brexit result, thinkbroadband.com said there is no immediate change being planned to any broadband schemes – including those in Wales.