Anscombe's injury meant Aaron Cruden, who is overdue a rest, had to wear the No10 jersey during last Friday night's one-point win over the Force after initially being listed in the reserves.
Rennie said he had looked to bring in a Kiwi player from overseas to replace Kahui, but was turned down by the New Zealand Rugby Union as it was after the April 1 cut-off.
"The expectation is we have to find someone in New Zealand to replace Richard and obviously we've already got Save [Tokula] in our ranks so the cupboard is pretty bare. If we lost another one we'd be under a bit of pressure," he said.
Rennie said Dan Waenga, a former Chiefs development player, would be called in to cover first-five in Anscombe's absence.
The presence of Bundee Aki, Ngatai's return and Tokula's solid performance against the Force in his sixth Super rugby appearance for the Chiefs in four years means Rennie has adequate cover for what is an important game against the Hurricanes at the Cake Tin on Friday.
It's bad luck for former wing or fullback Nanai-Williams, who has been outstanding for the Chiefs this season, his attacking ability in midfield making the All Black selectors, who have midfield issues of their own, take notice.
"He's certainly attacked really well," said Rennie.
"He's stepped up big time for us. We've had a bit of a crisis in midfield and he's played most of his rugby at No13 this year and done a good job."