Burke indicated during the final press conference that he would like to return to the post of GM with another club as soon as possible, but Gaglardi dismissed the possibility of him joining Dallas.
"Brian once said that when he takes something on there are two hands on the steering wheel and both of them are his," Gaglardi told CKNW radio, as quoted by the Vancouver Sun.
"He's a top dog kind of guy and one day he'll be back in the league as a general manager.
"But right now I don't know how or where he'd fit in with us if he wasn't in charge."
Burke remains within Maple Leaf Sports & Entertainment Ltd. as an advisor to President and COO Tom Anselmi, though he is no longer involved in the Maple Leafs' organization.