Twyford confirmed that there was an employment dispute between Barclay and the Ministry of Housing and Urban Development, and he hoped it would be resolved as soon as possible.
He denied a Herald report that it may revolve around the transfer of the KiwiBuild Unit from the Ministry of Business, Innovation and Employment to the new Ministry of Housing and Urban Development.
"There is no basis that I'm aware of for that assertion," Twyford said, although he acknowledged that he could not "unequivocally rule it out" because he had been briefed only on a very high level.
Both Twyford and Prime Minister Jacinda Ardern have declined to go into detail about Barclay's absence as chief executive of the Government's flagship KiwiBuild programme, just a few months into the role.
Twyford said today it was an unwelcome distraction but the KiwiBuild programme was much bigger than one person.
"There's an entire new ministry devoted to implementing this programme. It's much, much bigger than one person."
Barclay's absence comes at the same time as another chief executive has announced his resignation.
New Zealand Transport Agency head Fergus Gammie yesterday announced he had resigned.
It follows mounting pressure on Gammie and the agency over revelations that the NZTA had not been performing its regulatory function properly and as a result, thousands of vehicles were issued dodgy warrants of fitness.
Yesterday Twyford would not comment on Gammie's departure, with a spokeswoman saying it was an operational matter.
Today Twyford was happy to comment, saying Gammie had done the right thing.
"Mr Gammie has, I think, rightly concluded that he is in an untenable situation and he wants the work of the organisation and the current review that's underway to get the Transport Agency's regulatory role back on a sound footing without the distraction of any question marks about his role as chief executive."
National leader Simon Bridges said losing two chief executives from cornerstone government agencies was an "absolute shambles".
"I think what it shows is behind the scenes in both of these entities, they're core infrastructure entities for the Government, the level of dysfunction. You've got a situation in KiwiBuild, we're seeing every day the shambolic nature of it and that's led to a breakdown in the relationship, and you've got the same in NZTA," Bridges said.
He said it was too convenient that the Prime Minister and Twyford were able to say they could not comment.
"The public has a right in general terms to know what's happened here with two chief executive in relation to our two big infrastructure agencies."
National's housing spokeswoman Judith Collins said the secrecy around Barclay's absence made people more curious about what was going on.
"The head of KiwiBuild is clearly a major role. There was a lot of song and dance and celebrating when Mr Barclay was appointed.
"I would have thought that given the contractual obligations and the possible risk to the Crown and taxpayers, Phil Twyford needs to front up," she said.
National's transport spokesman Paul Goldsmith said more detail was also needed around Gammie's departure.
"What we don't have any clarity about in this case is whether he was pushed or not," Goldsmith told reporters.
