The Chiefs sought permission after injuries sidelined centers Andrew Horrell, Tim Nanai-Williams and All Blacks center Richard Kahui.
Horrell may miss the rest of the season with an ankle injury, Nanai-Williams was discharged from hospital last week after treatment for a stomach ulcer, and Kahui has a career-threatening shoulder injury.
The Chiefs' shortage of fit backs became more serious when fullback Gareth Anscombe broke a toe.
Coach Dave Rennie says the Chiefs asked permission to call in an overseas player but were refused "because there are apparently some rules around time frames."