The Scottish teenager has operated without a coach since April when he split with Mark Petchey, saying there were coaches better suited to carrying him into the world's top 10.
Former top 10 player Gilbert has all the credentials, having steered both Andre Agassi and Murray's Wimbledon victim Andy Roddick to the top of the rankings.
"It is not a done deal, but I've spoken to Brad," 19-year-old Murray told BBC radio.
"He's obviously someone that coached Agassi and Roddick and got both of them to number one in the world.
"He's got a very good record as a coach and he is somebody that I'd like to work with."
One stumbling block, however, would be that Gilbert, currently working for US broadcaster ESPN, has expressed reservations about being away from his family for long periods.