But after impressing there in his side's European semi-final victory over Newport Gwent Dragons, Burleigh is staking his claim for the shirt.
And with Tom Heathcote leaving BT Murrayfield at the end of the season, Burleigh is eager to take on the role.
"I guess I'm a utility," Burleigh told the BBC.
"I've got to perform in that position - I really enjoyed it (against the Dragons), I'd love to play there a bit more."
Edinburgh head coach Alan Solomons believes the Kiwi can prove a perfect fit in the role and admitted he was shocked Burleigh had never had much of an opportunity to play in the position in Super Rugby.
"Phil is a fantastic 10, and - I've spoken to him about this - I'm surprised in New Zealand they never played him at 10," said Solomons.
"I know they had a number of other good players playing, but Phil is the absolute New Zealand first and second five-eighth, he is an outstanding rugby player."
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