"We may not make June 30, but we will be close," the senior official told reporters as top US diplomat John Kerry prepared to head today for what could be the last talks between Iran and global powers on the deal.
The official said the target date to finalise the historic deal -- the main outlines were agreed in April in Lausanne, Switzerland -- would only "slip" by a few days.
Others, including Iranian Foreign Minister Mohammad Javad Zarif, who is due in Vienna tomorrow, have said the deadline may be missed by a few days but until now Washington had insisted it still aimed to get a deal on schedule.
"We can truly see a path forward that gets us to a very good agreement here. We know what the pieces of it are," the US official said, adding that in the end, Iran was facing "critical choices."
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Iran, which has engaged in something of a rapprochement with the West since the election of President Hassan Rouhani in 2013, denies wanting nuclear weapons, saying its nuclear programme is exclusively peaceful.
Under the Lausanne framework, which was also agreed several days late, Iran will downsize its nuclear activities in order to make any attempt to make a nuclear weapon all but impossible.
This includes Iran cutting the number of centrifuges enriching uranium, which is used in nuclear power but also for a bomb when highly purified, as well as slashing its uranium stockpiles and changing the design of a new reactor.